Mechanical arm including a counter-balance

ABSTRACT

A mechanical arm comprises a forearm, a spring, and an upper arm disposed between the forearm and the spring, wherein the forearm applies a moment to the upper arm. A copying device associated with the upper arm copies a force associated with the moment to the spring. The spring is adapted to apply a counter-force resisting at least a portion of the moment to reduce a torque for urging the forearm.

CLAIM OF PRIORITY

This application is a continuation of U.S. patent application Ser. No.10/443,459, entitled “Counter Balance System and Method with One or MoreMechanical Arms,” filed May 22, 2003, now U.S. Pat. No. 7,428,855 whichclaims priority to U.S. Provisional Application No. 60/382,497, filedMay 22, 2002 and U.S. Provisional Application No. 60/382,654, filed May23, 2002, both of which are incorporated herein in their entireties.

FIELD OF THE INVENTION

The present invention is directed to mechanical arm systems includingone or more mechanical arms.

BACKGROUND OF THE INVENTION

There are a number of robotics systems including one or multiple armswhich are linked together in order to perform tasks such as lifting andmoving objects and tools from one location to another in order toperform these tasks. As the arms and objects and tools have weight,other substantial motors must be used in order to move packages from onelocation to another. With such motors, such systems may not be as userfriendly as desirable. In other words, such systems may requiresubstantial energy in order to operate and will not have as delicate atouch as required for various situations.

SUMMARY OF THE INVENTION

The invention is directed to overcome the disadvantages of prior art.The invention includes a number of features which are outlined below.

An Adjustable Counterbalance with Counterbalanced Adjustment (RotaryJoints)

A system is presented for counterbalancing the gravitational moment on alink when the link is supported at a point. A first spring mechanismbalances the link about all axes that pass through the support point.The link can be balanced throughout a large range of motion. When loadis added to or removed from the link, the first spring mechanism can beadjusted to bring the link back into balance. The force that is requiredto adjust the first spring mechanism is counterbalanced by a secondmechanism with one or more additional springs. Little external energy isneeded to adjust the counterbalance for a new load. Little externalenergy is needed to hold the load or to rotate the link and load to anew position. Unlike counterweight based balance systems, the springsystem adds little to the inertia and weight to the link. The system canbe adjusted to deliver a moment that does not balance the link. Theresulting net moment on the link can be used to exert a moment or forceon an external body.

A Counterbalance System for Serial Link Arms

Several systems are presented for counterbalancing mechanical arms thathave two or more links in series. The joints between the links may haveany number of rotational degrees-of-freedom as long as all of the axesof rotation pass through a common point. For each distal link that hasany vertical motion of its center-of-gravity, a series of one or morepantograph mechanisms are coupled to the link. The motion of the distallink is reproduced by the pantograph mechanisms at a proximal link wherea vertical orientation is maintained. A counterbalance mechanism isattached to the proximal end of the chain of pantograph mechanisms. Theproximal location of the counterbalance minimizes the rotational inertiaof the arm. The counterbalance torque couples only to the balanced link.Spring or counterweight balance mechanisms can be used. A pantographmechanism can also be used to move the counterbalance to a locationwhere space is available.

An Adjustable Counterbalance with Counterbalanced Adjustment(Translational Joints)

A system is presented for counterbalancing the gravitational force on alink when the link is constrained by a prismatic joint to translatealong a linear path. An extension spring with a stiffness K is connectedto the link. A second spring mechanism with a stiffness of negative K isalso connected to the link. As the link translates, the net spring forceon the link is constant. The net spring force can be changed byadjusting the pretension on either spring. The force that's required toadjust the pretension is counterbalanced by a third mechanism with oneor more additional springs. When load is added to or removed from thelink, or when the slope of the prismatic joint is changed, the systemcan be adjusted to rebalance the link. Little external energy is neededto adjust the counterbalance for a new load. Little external energy isneeded to hold the load or to move the link and load to a new position.Unlike counterweight based balance systems, the spring system addslittle to the inertia and weight to the link. The system can be adjustedto deliver a force that does not balance the link. The resulting netforce on the link can be used to exert a force on an external body. Thesystem can be converted to counterbalance rotational motion byconnecting the link to a Scotch Yoke mechanism.

Multiple Counterbalance Mechanisms Coupled to One Axis of Rotation

Two or more counterbalance mechanisms can be coupled to one axis ofrotation. The net sinusoidal torque phase and magnitude can be changedby adjusting the magnitude or phase of the individual mechanisms. Withmultiple counterbalance mechanisms, a wider dynamic range of loads canbe balanced. With the ability to adjust the phase of the sinusoidaltorque, the system can be used to exert a reaction force in an arbitrarydirection on another body. Embodiments of the invention further include:

-   1. An adjustable load energy conserving counterbalance mechanism and    method as shown in the attached figures.-   2. A multiple series link balance mechanisms and methods as shown in    the attached figures.-   3. An adjustable counterbalance with a counterbalance adjustment    with rotary joints as shown in the figures.-   4. A counterbalance system for series link arms as shown in the    figures.-   5. An adjustable counterbalance and counterbalance adjustment with    translational joints as shown in the figures.-   6. Multiple counterbalance mechanisms and methods coupled to one    axis of rotation shown in the figures.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an illustration of a gravity counterbalance free body;

FIG. 2 is a graphical illustration of the helical spring forcedeflection curves;

FIG. 3 is an illustration of the rotary link gravity counterbalancemechanism;

FIGS. 4 a and 4 b are illustrations of the cable gimbal counterbalance;

FIGS. 5 a and 5 b are illustrations of the spiral spring mechanism;

FIGS. 6 a, 6 b, and 6 c are illustrations of the two degree of freedomcable gimbal mechanism;

FIGS. 7 a, 7 b, and 7 c are illustrations of the two degree of freedomcable gimbal with meshing pulleys;

FIGS. 8 a, 8 b, and 8 c are illustrations of the two degree of freedomcable gimbal mechanism;

FIGS. 9 a and 9 b are illustrations of the manual and motorizedadjustment mechanisms;

FIG. 10 is an illustration of a free body diagram of the counterbalanceadjustment force;

FIGS. 11 a and 11 b are graphical illustrations of the adjusting forcecurves;

FIGS. 12 a and 12 b are illustrations of the load adjustmentcounterbalance;

FIG. 13 is a free body diagram of a sliding pivot spiral pulley;

FIGS. 14 a, 14 b, and 14 c are spreadsheets of equations for a slidingpivot spiral pulley;

FIG. 15 is a graphical illustration of a pulley tangent radius forsliding pivot, constant torque, spiral pulley;

FIG. 16 is an illustration of a sliding pivot, constant torque, spiralpulley;

FIG. 17 is a graphical illustration of a pulley tangent radius for asliding pivot, parabolic torque, spiral pulley;

FIG. 18 is an illustration of a sliding pivot, constant and parabolictorque, spiral pulleys;

FIG. 19 is a free body diagram of a fixed pivot spiral pulley;

FIGS. 20 a and 20 b are spreadsheets of questions for a fixed pivotspiral pulley;

FIG. 21 is a graphical illustration of a pulley tangent radius for afixed pivot, parabolic torque, spiral pulley;

FIG. 22 is a graphical illustration of a comparison of fixed pivot andsliding pivot pulley radii;

FIGS. 23 a and 23 b are illustrations of a load adjustmentcounterbalance with the spiral pulleys on the carriage;

FIGS. 24 a and 24 b are illustrations of a manual link-anglecompensation;

FIGS. 25 a and 25 b are illustrations of a link-angle compensationcounterbalance;

FIGS. 26 a and 26 b are illustrations of a rotary link-anglecompensation counterbalance;

FIGS. 27 a and 27 b are illustrations of a simplified link-anglecompensation counterbalance;

FIGS. 28 a and 28 b are illustrations of a load compensationcounterbalance;

FIGS. 29 a and 29 b are illustrations of the external and internal camand roller;

FIGS. 30 a and 30 b are illustrations of a dual opposed counterbalancemechanism;

FIGS. 31 a and 31 b are illustrations of the multiple opposedcounterbalance mechanisms;

FIGS. 32 a and 32 b are illustrations of the dual phase shiftedcounterbalance mechanism;

FIGS. 33 a, 33 b, and 33 c are illustrations of the translationalcounterbalance force diagram;

FIGS. 34 a and 34 b are illustrations of the adjustable constant forceor constant torque mechanism;

FIGS. 35 a and 35 b, 36 a and 36 b, and 37 a and 37 b are illustrationsof the adjustable translational counterbalance;

FIGS. 38 a and 38 b, and 39 a and 39 b are illustrations of thetranslational counterbalance with an adjustment counterbalance;

FIGS. 40 a and 40 b are illustrations of a translational counterbalancewith an adjustment counterbalance and position compensation;

FIGS. 41 a and 41 b are illustrations of a rotary counterbalance with ascotch yoke;

FIGS. 42 a and 42 b are illustrations of a universal joint;

FIG. 43 is an illustration of a phase shifted counterbalance mechanism;

FIG. 44 is an illustration of a two degree of freedom elbow pantograph;

FIGS. 45 a, 45 b and 45 c are illustrations of a three degree of freedomelbow pantograph;

FIGS. 46 a and 46 b are illustrations of a remote rotary counterbalancewith a scotch yoke;

FIGS. 47 a, 47 b, and 47 c are illustrations of a pitch axis elbowjoint;

FIGS. 48 a, 48 b, and 48 c are illustrations of an arm with a roll andpitch axis elbow joint;

FIGS. 49 a, 49 b, and 49 c are illustrations of an arm with a large ROMelbow and shoulder;

FIG. 50 is an illustration of an arm with a pitch and yaw axis elbowjoint;

FIG. 51 is an illustration of an arm with a three DOF elbow joint;

FIGS. 52 a, 52 b, and 52 c are illustrations of an arm with a three DOFshoulder joint;

FIGS. 53 a, 53 b, and 53 c are illustrations of an arm with a four DOFshoulder joint;

FIGS. 54 a, 54 b, and 54 c are illustrations of an arm with a two DOFshoulder and two DOF elbow; and

FIGS. 55 a and 55 b are illustrations of an adjustable stiffness rotarycounterbalance.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS Outline of Theory ofOperation

-   1. General Case, Rotational Gravity Counterbalance-   2. Zero-Length Spring and Cable Gimbal Mechanism-   3. Adjustment of the Gravity Counterbalance    -   Analysis of the force needed to adjust the gravity        counterbalance-   4. Counterbalancing of the Adjustment Mechanism    -   The required force profile    -   Derivation of the geometry for a sliding-pivot spiral pulley    -   Derivation of the geometry for a fixed-pivot spiral pulley-   5. Link-Angle Compensation and Counterbalance Mechanism    -   Other versions of the link-angle compensation and counterbalance        mechanism-   6. Load Compensation and Counterbalance Mechanism-   7. System Operation    -   Fixed Gravity Counterbalance    -   Adjustable Gravity Counterbalance    -   Gravity Counterbalance with Counterbalanced Adjustment    -   Gravity Counterbalance with Counterbalanced Adjustment and        Link-Angle Compensation    -   Gravity Counterbalance with Counterbalanced Adjustment and        Link-Angle Compensation with Counterbalance    -   Gravity Counterbalance with Counterbalanced Adjustment,        Link-Angle Compensation with Counterbalance, and Load        Compensation    -   Gravity Counterbalance with Counterbalanced Adjustment,        Link-Angle Compensation with Counterbalance, and Load        Compensation with Counterbalance-   8. Multiple Counterbalance Mechanisms on the same Axis of Rotation    -   Dual Opposed Counterbalance    -   Multiple Mechanisms for adjustable phase and magnitude    -   Dual Phase Shifted Counterbalance Mechanism-   9. Translational Counterbalance Mechanisms-   10. A Rotary Counterbalance made from a Scotch Yoke and    Translational Counterbalance Mechanism-   11. Extending the Counterbalance to Multiple Degrees of Freedom-   12. Extending the Counterbalance to Multiple Link Arms    -   Pantograph Mechanisms    -   Reasons for using a Pantograph Mechanism    -   Examples of Pantograph Mechanisms    -   Axial Offset Pantograph    -   Phase Shifting Pantograph    -   One or Two Degree of Freedom Pantograph    -   Three Degree of Freedom Pantograph    -   Other Parallel Axis Pantograph Mechanisms    -   Pantograph Mechanisms in Series-   13. Examples of Counterbalanced Two Link Arms    -   Mounting constraints for the pantograph axis        A System that Uses Weight to Store Energy        The Effect of System Efficiency on Energy Consumption        Theory of Operation        1. General Case, Rotational Gravity Counterbalance

The two-link arm has two gravity counterbalance mechanisms, one for eachlink. One of the mechanisms provides torque to counterbalance thegravity moment at the shoulder joint. The other mechanism providestorque to counterbalance the gravity moment at the elbow joint. Let'sfirst develop the equations for counterbalancing one link at a time.

An unconstrained rigid body has a total of 6 degrees of freedom or DOF.Three of the DOF are translational, and three are rotational. FIG. 1shows a free body diagram of a rigid body or link. The link has a massM, with its center of gravity located at a point D. The link issupported in all three translational DOF at a point C, located adistance L from point D. A point B is located on the vertical line thatpasses through point C. The angle between CB and CD is θ.

The weight of the link will act at the center of gravity of the linkwith a downward force f=Mg. There is a horizontal line passing throughpoint C, that is perpendicular to both line CD and line CB. The momentor torque T₁ about the horizontal line, exerted by gravity force f is:T₁=MgL sin θ  eq. 1

Point A is located on the line determined by points C and D. Thedistance between points A and B is c. The distance between points A andC is b. The distance between points B and C is a. The angle between ABand AC is φ.

To counterbalance the gravity torque T₁, a spring or other mechanismdelivers a tension force F between points A and B. Force F isproportional to a constant K₁ and distance c. F can be expressed by thefollowing equation:F=K₁c.  eq. 2

The torque T₂ exerted on the link about the horizontal line by force FisT ₂ =−Fb sin φ  eq. 3

From the law of sines for plane triangles,

$\begin{matrix}{\frac{a}{\sin\;\varphi} = \frac{c}{\sin\;\theta}} & {{eq}.\mspace{14mu} 4}\end{matrix}$

Solving for sin φ in equation 4, and substituting into equation 3 yields

$T_{2} = {- \frac{{Fba}\;\sin\;\theta}{c}}$

Substituting K₁c for F in equation 4 yieldsT ₂ =−abK ₁ sin θ  eq. 5

The link will be counterbalanced when the sum of the moments about thehorizontal line is zero.or T ₁ +T ₂=0

Substituting equation 1 and equation 5 for T₁ and T₂ yields:

$\begin{matrix}{\;{{{{M\; g\; L\mspace{14mu}\sin\mspace{11mu}\theta} - {a\; b\; K_{1}\mspace{11mu}\sin\mspace{11mu}\theta}} = {0\mspace{20mu}{Or}}}{{M\; g\; L} = {a\; b\; K_{1}}}}} & {{eq}.\mspace{14mu} 6}\end{matrix}$

In other words, with a spring or other mechanism that satisfies equation2 above, the link can be counterbalanced. It is important to note thatthe above derivation does not place any limit on the rotational degreesof freedom of the link. The only constraint is that the translationaldegrees of freedom are fixed at point C. The horizontal line or axisabout which the moments were calculated is a theoretical construct. Aphysical pivot joint with the horizontal line as its axis of rotation isnot needed. In fact, even if all three rotational DOF of the link werefixed at point C, the moments exerted on the link by these constraintswould be zero.

With equation 6 satisfied, the link will be counterbalanced about anyand all axes that pass through point C. The link will be balanced in anyorientation, over the entire range of motion or ROM of all threerotational degrees of freedom. Dimensions a, b, and spring constant K₁can be positive or negative.

2. Zero-Length Spring and Cable Gimbal Mechanisms

From equation #2 above, the force that is required to counterbalance alink is proportional to the distance between points A and B. A springthat meets this requirement is sometimes called a “zero-length” spring.The term “zero-length” does not mean that the spring has a dimension ofzero, but that the force extrapolates to zero as the distance betweenthe spring pivots goes to zero.

FIG. 2 shows force-deflection curves for conventional helical extensionsprings. A large initial tension is needed for a spring to meet thezero-length criteria. An analysis of the torsional stress produced bythe initial tension shows that zero-length springs fall outside of therange that is recommended by the Spring Manufacturers Institute.

A real spring will have tolerances associated with its force-deflectioncurve. Extension springs are usually specified by their spring constantor stiffness, the initial tension, and the distance between end hooks.If a spring is specified to have a force of zero at “zero-length”, therewill be a tolerance associated with the actual force at zero length.

FIG. 3 illustrates another problem with using a conventional extensionspring to counterbalance a link. In FIG. 3, an extension spring ismounted on two pivots. The axis of one pivot passes through point A, andthe axis of the other pivot passes through point B. When the link angleθ=0°, distance c becomes: c=a−b. If the minimum length between springpivots is greater than (a−b), then the spring will “bottom out” beforethe link can reach θ=0°. The angular range of motion of the link will berestricted by the spring.

Increasing dimension a, or decreasing dimension b, will make room forthe spring. This will enable the link to rotate throughout a full 360°.This has a drawback however. As (a−b) gets larger, the energy stored inthe spring gets larger too. As a result, for a given load, a largerspring is needed. The energy stored in a zero-length spring is equal to:E=½K₁c²

When the link angle θ=0°, and c=a−b, the energy stored in the springwill be:E _(min)=½K ₁(a−b)²

When the link rotates to θ=180°, the spring must store the gravitationalpotential energy of the link. This is equal to:E_(grav)=2MgL

The total energy stored in the spring at q=180°, is equal to:E _(total)=½K ₁(a−b)²+2MgL

This shows that as (a−b) gets larger, a larger spring is needed tocounterbalance the same maximum load.

FIGS. 4 a and 4 b show a link that is counterbalanced by a spring. Thespring is remotely located from points A and B. The force from thespring is transmitted to the link by a flexible cable. This can becalled a “cable gimbal mechanism”. At one end, the cable is attached tothe spring. The cable extends from the spring, along an axis that passesthrough point B. This axis is parallel to the axis of rotation of thelink. The cable then wraps 90° around an idler pulley. The far end ofthe cable is attached to a pivot bearing that is attached to the link.The pivot bearing rotates about an axis that passes through point A. Thepivot axis is also parallel to the axis of rotation of the link. Theidler pulley is mounted in a yoke. The yoke is mounted on bearings in aframe or carriage that holds the spring. The yoke bearings rotate aboutthe previously defined axis passing through point B. The cable passesthrough a hollow shaft in the pulley yoke.

The cable delivers a force that acts between the axis passing throughpoint A and the axis passing through point B. The force on the link isthe same as if the spring were located between the two axes. This “cablegimbal mechanism” allows dimension a to be very small without the springrestricting the rotation of the link. The cable gimbal mechanism can beadjusted to balance a link over a wide range of loads.

FIGS. 4 a and 4 b show how the fixed end of the spring can be attachedto the frame. The spring is hooked onto a threaded spring anchor, U.S.Pat. No. 4,032,129, available from Century Spring Corp. in Los Angeles,Calif. A nut is threaded onto the screw anchor to hold it to the frame.The screw anchor and nut can be used to adjust the spring tension. Thetension can be adjusted so that the force is proportional to distance c.A special “zero-length” spring is not needed for the cable gimbalmechanism. The tolerances on the spring can be very loose withoutaffecting the performance of the system.

Other types of springs can be used in place of the extension spring.FIGS. 5 a and 5 b show a spiral spring mechanism. The outer end of thespiral spring is attached to a frame. The inner end of the spring isattached to a shaft. The shaft is mounted to the frame on a pair of ballbearings. A constant diameter capstan is rigidly attached to the shaft.A cable gimbal mechanism is attached to the frame. One end of a flexiblecable is attached to the capstan. The cable wraps around the capstan andthen passes through the cable gimbal. When the cable is extended, thetension will increase linearly with deflection. A helical torsion springor a torsion bar spring can be used in place of the spiral spring.

FIGS. 6, 7, 8, and 13 in U.S. Pat. No. 5,435,515 show several otherspring assemblies that can be used in place of the helical extensionspring.

The cable gimbal shown in FIGS. 4 a and 4 b has one degree of freedom.The yoke and idler pulley are free to rotate about axis B. FIGS. 6 a-6c, 7 a-7 c, and 8 a-8 c, show cable gimbal mechanisms with two degreesof freedom. A two degree of freedom mechanism is often needed tocounterbalance a link with more than one degree of freedom.

The cable gimbal shown in FIGS. 6 a, 6 b and 6 c has two yokes. An idlerpulley is mounted to each yoke. The smaller of the two yokes is mountedon a pair of bearings in the larger yoke. The bearings enable the yoketo pivot about axis #1. The larger yoke is mounted on another pair ofbearings. These bearings enable the outer yoke to pivot about axis #2.Axis #1 and axis #2 intersect at a point. The counterbalance cableenters the gimbal along axis #2. The cable wraps part way around theidler pulley on the larger yoke. It then transfers over to the idlerpulley on the smaller yoke. The cable wraps 90° around the second idlerpulley. At that point, the cable leaves the cable gimbal assembly. Thefar end of the cable is attached to a u-joint that is attached to thecounterbalanced link. The intersection of axis #1 and axis #2corresponds to point B in FIG. 1. The intersection of the axes ofrotation of the u-joint corresponds to point A in FIG. 1. The cabletension acts along a line that passes through points A and B.

FIG. 7 shows another two degree of freedom cable gimbal mechanism. Thegimbal yoke is pivoted about a pair of bearings. The cable enters theyoke along the pivot axis. Three idler pulleys and two guide rollers aremounted to the yoke. Two of the idler pulleys may have gear teeth cutinto their periphery. These two idler pulleys are mounted so that theiraxes of rotation intersect the yoke pivot axis. The geared pulleys haveparallel axes. The axis of the third idler pulley is parallel to theaxes of the first two. All three of the idler pulleys are bisected bythe plane containing the yoke gimbal axis. The third idler pulley holdsthe cable to the pivot axis as the cable enters the yoke assembly. Thecable wraps part of the way around one of the geared pulleys. The cablepasses between the two geared pulleys. The cable passes between twoguide rollers as it exits the gimbal assembly. The guide rollers holdthe cable so that it stays in the plane that bisects the gimbalassembly. The far end of the cable is attached to a u-joint that isattached to the counterbalanced link. The cable tension acts in theplane that bisects the gimbal assembly. The position of point A movesslightly as the cable direction changes. Point A is located in thebisecting plane, at or near the midpoint between the two geared pulleyaxes.

This cable gimbal should also work with only one of the two gearedpulleys. The position of point A will move more through. The cablegimbal should also work without the gear teeth.

The cable gimbal mechanism in FIGS. 8 a, 8 b and 8 c is similar to themechanism in FIGS. 6 a, 6 b and 6 c. An additional idler pulley has beenadded to the outer gimbal. This enables the gimbal to operate over alarge range of motion without mechanical interference. The cable entersthe outer gimbal through a hollow shaft. In the outer gimbal, the cablewraps around two idler pulleys. From there, the cable wraps one quarterof a turn around the idler pulley on the inner gimbal. The cable passesthrough a plastic guide as it exits the inner gimbal.

The outer gimbal is mounted on a pair of bearings. At the hollow shaftend, a ball bearing is used to take the thrust and radial loads. Aneedle bearing is used at the other end. The inner gimbal is alsosupported by a ball bearing and a needle bearing.

3. Adjustment of the Gravity Counterbalance

Now lets look at how the counterbalance mechanism can be adjusted tobalance a link if MgL of the link changes. MgL may change if the mass Mchanges, if the distance L to the center of gravity changes, or even ifthe local gravitational acceleration g changes. To maintain balance,equation 6 must be satisfied. Balance can be maintained by changing anycombination of the parameters a, b, or K₁. As will be seen later, oneparticularly useful way of adjusting the gravity counterbalancemechanism is to adjust dimension a. The mechanism to adjust dimension acan be located on the adjacent link or on ground. This helps to reducethe mass, size, and inertia of the moving link.

FIGS. 9 a and 9 b show different mechanisms for adjusting dimension a.In both figures, the spring carriage is guided by a linear bearing. Thespring carriage is constrained by a bearing to move in a verticaldirection. In FIG. 9 a, a leadscrew is attached to the spring carriage.The leadscrew passes through a fixed member. A nut on the far end of theleadscrew can be turned to move the spring carriage up or down.

FIG. 9 b shows how the nut can be replaced with a nut and gear. A motorand pinion, can then be used to power the adjustment of dimension a. Anencoder and brake may be added along with a control system to automatethe adjustment.

Next, we will look at the force that's required to adjust dimension a.FIG. 10 shows a free body diagram of a counterbalanced link. As before,force F=K₁c. In other words, force F is proportional to c, the distancebetween points A and B. If we define F₁ as the vertical component offorce F, then F₁ will be proportional to the distance d, orF₁=K₁d

The distance between points C and D is −b cos θ. As a result,d=a−b cos θor, the adjusting force is: F ₁ =K ₁(a−b cos θ)  eq. 7

From equation 7, it can be seen that under most conditions, F₁ ? 0 As aresult, when dimension a is adjusted, energy is transferred into or outof the gravity counterbalance spring. The next section will show howforce F₁ can be counterbalanced with the force from another springmechanism.

4. Counterbalancing of the Adjustment Mechanism the Required ForceProfile

A force is required to adjust each gravity counterbalance mechanism. Theadjusting force can be counterbalanced with a second mechanism. Thissecond mechanism is called the adjustment counterbalance mechanism.

From equation 7, the force required to adjust a gravity counterbalancemechanism varies with the spring constant K₁ of the gravitycounterbalance spring, dimension a, dimension b, and the link angle θ.Assuming that K₁, b, and θ, are constants, I will first show how aspring mechanism can be used to balance force F₁, as only dimension achanges. Later, I will show how the mechanism can be modified to balancethe adjustment force when θ or b are changed. First, lets call thespring that counterbalances the gravity torque on a link, spring #1. Asspring #1 extends, the adjustment counterbalance spring systemcontracts. From equation 7, FIG. 11 a shows a graph of F₁. This is theforce required to extend spring #1 as a function of dimension a. FIG. 11b shows a graph of the force required to extend the adjustmentcounterbalance spring mechanism as a function of dimension p. Dimensionp is the extension of the counterbalance spring mechanism. The twofigures represent the same force curve as viewed from differentreference frames. Remember that when spring #1 extends, spring #2contracts. The coordinate transformation from FIG. 11 a to FIG. 11 b isproduced by substituting a=(e−p) into equation 7 to yield the followingequation for the adjustment counterbalance spring system.F′ ₁ =−K ₁ p+K ₁(e−b cos θ)  eq. 8

The curves in FIG. 11 b represent the force-deflection characteristicsrequired from the adjustment counterbalance spring system. It has anegative spring constant −K₁. The force decreases as the spring extends.No common spring has this type of force-deflection curve. As a result, aspecial spring mechanism is needed for the adjustment counterbalance.

FIGS. 12 a and 12 b show a front and side view of a spring mechanismthat will produce the negative stiffness characteristics required tobalance the adjustment force. The gravity counterbalance spring is shownas spring #1. As before, spring #1 is attached to a frame or carriagethat is guided in a vertical direction. A pivot is attached to thecarriage. One end of a flexible cable is attached to the pivot. Thecable wraps around a spiral pulley. The far end of the cable is attachedto the pulley.

A second spring, spring #2, is a helical extension spring with astiffness of K₂. One end of spring #2 is attached to a fixed-pivot. Asecond flexible cable is attached to the other end of spring #2. Thesecond cable wraps around a second spiral pulley. The end of the secondcable is fixed to the second pulley.

The two spiral pulleys are coupled, either directly, through a driveshaft, or by other means. The pulleys are mounted on low frictionbearings.

The force from spring #1, acting on the first spiral pulley, produces atorque in a clockwise direction. The force from spring #2, acting on thesecond spiral pulley, produces a torque in a counter-clockwisedirection. If the magnitudes of the torques are equal, then the nettorque is zero. With zero net torque, the force from spring #1 iscounterbalanced.

The two spiral pulleys can be designed so that the net torque is zeroover a range of pulley rotation. The force from spring #1 will becounterbalanced over this range of rotation. Note that the torque foreach pulley does not need to be constant throughout the rotation of thepulleys. For balance, the torques need only be equal to each otherthroughout the range of rotation.

The two spiral pulleys and two cables make up a transmission. Whenviewed from the reference frame of the cable attached to spring #1, thereduction ratio varies continuously from a very high ratio to a very lowratio. The transmission converts the positive stiffness force fromspring #2 into a negative stiffness force. It should be possible tosubstitute either spiral pulley with a cam and a roller follower. Thespring can be connected to the roller follower. Pulleys and cables alongwith cams and roller followers both have very high mechanicalefficiency.

Derivation of the Geometry for a Sliding-Pivot Spiral Pulley

The two spiral pulleys are not the same. Spiral pulley #1 is connectedby cable to a pivot. The pivot slides along a linear path. Spiral pulley#2 is connected by cable directly to spring #2. The opposite end ofspring #2 is attached to a fixed-pivot. Spiral pulley #1 can be referredto as a “sliding-pivot” pulley. Spiral pulley #2 can be referred to as a“fixed-pivot” pulley.

The shape of each pulley can be determined by the solution of adifferential equation. Let's first look at pulley #1. FIG. 13 shows afree body diagram of a spiral pulley with a sliding-pivot. The axis ofrotation of the cable pivot passes through point P. The axis of rotationof the pulley is parallel to the cable pivot axis. The pulley axispasses through point Q. Point P translates along a linear path that is adistance h from the pulley axis. The flexible cable extends from thepivot at point P to a tangent on the spiral pulley. The cable is in theplane that is normal to the axis of rotation of the pulley. Dimension ris the tangent radius or perpendicular distance from the pulley axis tothe flexible cable. The distance along the flexible cable from point Pto radius r is dimension z. Dimension p is the component of the distancebetween points P and Q in the direction of the translation of point P.

Angle a is the angle of rotation of the spiral pulley. Angle b is theangle between the flexible cable and the linear path of point P. Angle gis the angle between the linear path of point P and the line throughpoints P and Q. From equation 7, the force from spring #1, acting in thedirection of the sliding-pivot is:F ₁ =K ₁(a−b cos θ)If we define a distance x as: x=(a−b cos θ)  eq. 9then: F₁=K₁x  eq. 10

If F is the tension in the flexible cable, the component of this tensionin the direction of the translation of point P will be equal to F cos β.The system will be in balance when:F₁=F cos β.  eq. 11

Substituting equation 10 into equation 11 and solving for F:

$F = \frac{K_{1}x}{\cos\;\beta}$

The torque τ on the pulley, produced by the force F will be:

$\begin{matrix}{{\tau = {{Fr} = \frac{K_{1}{xr}}{\cos\;\beta}}}{{Solving}\mspace{14mu}{for}\mspace{14mu} r\text{:}}} & \; \\{r = \frac{\tau\;\cos\;\beta}{K_{1}x}} & {{eq}.\mspace{14mu} 12} \\{{solving}\mspace{14mu}{for}\mspace{14mu} a\text{:}} & \; \\{a = {\frac{\tau\;\cos\;\beta}{K_{1}r} + {\beta\;\cos\;\theta}}} & {{eq}.\mspace{14mu} 13}\end{matrix}$

The line segment from point P to point Q is the hypotenuse of twotriangles. Using the Pythagorean equation for both triangles:r ² +z ² =p ² +h ²Solving for z: z=(p ² +h ² −r ²)^(1/2)  eq. 14Solving for p: p=(r ² +z ² −h ²)^(1/2)  eq. 15

Now we need an equation for angle β.

$\begin{matrix}{{g = {\tan^{- 1}\left( {h/p} \right)}}{\left( {g + \beta} \right) = {\tan^{- 1}\left( {r/z} \right)}}{{{Solving}\mspace{14mu}{for}\mspace{14mu}{\beta:\;\beta}} = {{\tan^{- 1}\left( {r/z} \right)} - {\tan^{- 1}\left( {h/p} \right)}}}} & {{eq}.\mspace{14mu} 16}\end{matrix}$

When the pulley rotates by dα and angle β changes by dβ, then dimensionz will change by dz.dz=−r dα−r dβ  eq. 17

Equation 17 can be solved numerically with a computer by converting thedifferential equation into a finite difference equation. To do this,

d z  becomes  Δ z = z_(n) − z_(n − 1)d α  becomes  Δα = α_(n) − α_(n − 1)d β  becomes  Δα = β_(n) − β_(n − 1)          n      α_(n)  becomes?α            0

Substituting into equation 17:

$\begin{matrix}{{{z_{n} - z_{n - 1}} = {{- {r_{n - 1}\left( {{\Delta\alpha} + \beta_{n} - \beta_{n - 1}} \right)}}\mspace{14mu}{or}}}\text{}{z_{n} = {z_{n - 1} - {r_{n - 1}\left( {{\Delta\alpha} + \beta_{n} - \beta_{n - 1}} \right)}}}} & {{eq}.\mspace{14mu} 18}\end{matrix}$

The term β_(n)−β_((n-1)), is needed for the numerical solution to thedifferential equation, but β_(n) is not available when it is needed. Forsmall steps in the numerical solution,

As a result,

$\begin{matrix}{{{\beta_{n} - \beta_{n - 1}} = {\beta_{n - 1} - \beta_{n - 2}}}{{{As}\mspace{14mu} a\mspace{14mu}{result}},}} & \; \\{z_{n} = {z_{n - 1} - {r_{n - 1}\left( {{\Delta\alpha} + \beta_{n} - \beta_{n - 2}} \right)}}} & {{eq}.\mspace{14mu} 19}\end{matrix}$

Assuming the following values for initial conditions or constants:

α₀ = 0° starting pulley angle Δα = Step angle for numerical solution θ =Constant link angle r₀ = Starting pulley tangent radius b = Constantdistance from link axis of rotation to force application p₀ = Startingdistance from cable pivot to link axis h = Constant offset distance fromthe linear slide to the link axis of rotation K₁ = Spring constant τ =Either a constant or a desired torque profile as a function of α.

The following initial conditions can be calculated:z ₀=(p ₀ ² +h ² −r ₀ ²)^(1/2) from equation 14β₀=tan⁻¹(r ₀ /z ₀)−tan⁻¹ (h/p ₀) from equation 16 and z₀ above

$a_{0} = {\frac{t_{0}\cos\;\beta_{0}}{K_{1}r_{0}} + {\beta\;\cos\;\theta}}$from equation 13 and β₀ abovep ₀=(r ₀ ² +z ₀ ² −h ²)^(1/2) from equation 15From equation 9, x=(a−b cos θ)thus: (p+x)=p+(a−b cos θ)

Looking at the definitions of dimension p and dimension a as seen inFIG. 12, both a and p are in the same direction. As a gets longer, pgets shorter by the same amount. If we assume for now that (β cos θ) isa constant, then (p+x) will be constant. We can now solve for (p+x).

$\quad\begin{matrix}{\left( {p + x} \right) = \left( {p_{0} + x_{0}} \right)} \\{= {p_{0} + \left( {a_{0} - {\beta\;\cos\;\theta}} \right)}} \\{= {p_{0} + \frac{t_{0}\cos\;\beta_{0}}{K_{1}r_{0}} + {\beta\;\cos\;\theta} - {\beta\;\cos\;\theta}}}\end{matrix}$

Substituting for p₀ from above:

$\begin{matrix}{\left( {p + x} \right) = {\left( {r_{0}^{2} + z_{0}^{2} - h^{2}} \right)^{1/2} + \frac{t_{0}\cos\;\beta_{0}}{K_{1}r_{0}}}} & {{eq}.\mspace{14mu} 20}\end{matrix}$

From equation 12

$\quad\begin{matrix}{r = \frac{\tau\;\cos\;\beta}{K_{1}x}} \\{= \frac{\tau\;\cos\;\beta}{K_{1}\left( {p + x - p} \right)}}\end{matrix}$

Substituting equation 15 for

${p:}\; = \frac{\tau\;\cos\;\beta}{K_{1}\left\lbrack {\left( {p + x} \right) - \left( {r^{2} + z^{2} - h^{2}} \right)^{1/2}} \right\rbrack}$

Substituting equation 16 for β:

$\begin{matrix}{r = \frac{\tau\;{\cos\left( {{\tan^{- 1}\left( {r/z} \right)} - {\tan^{- 1}\left( {h/p} \right)}} \right)}}{K_{1}\left\lbrack {\left( {p + x} \right) - \left( {r^{2} + z^{2} - h^{2}} \right)^{1/2}} \right\rbrack}} & {{eq}.\mspace{14mu} 21}\end{matrix}$

At step n of the finite difference equation:

$\begin{matrix}{r_{n} = \frac{\tau_{n}\;{\cos\left( {{\tan^{- 1}\left( {r_{n}/z_{n}} \right)} - {\tan^{- 1}\left( {h/p_{n}} \right)}} \right)}}{K_{1}\left\lbrack {\left( {p + x} \right) - \left( {r_{n}^{2} + z_{n}^{2} - h^{2}} \right)^{1/2}} \right\rbrack}} & {{eq}.\mspace{14mu} 22}\end{matrix}$

For small step size Δα, β_(n)≈β_(n-1). When angle β is less than 20°,cos β_(n)≈ cos β_(n-1) is a very good approximation. As a result:

$\begin{matrix}{r_{n} \cong \frac{t_{n}\;{\cos\left( {{\tan^{- 1}\left( {r_{n - 1}/z_{n - 1}} \right)} - {\tan^{- 1}\left( {h/p_{n - 1}} \right)}} \right)}}{K_{1}\left\lbrack {\left( {p + x} \right) - \left( {r_{n}^{2} + z_{n}^{2} - h^{2}} \right)^{1/2}} \right\rbrack}} & {{eq}.\mspace{14mu} 23}\end{matrix}$

Equation 23 has r_(n) on both sides of the equation. If the othervariables are known, r_(n) can be solved for by numerical methods.Microsoft Excel has an equation solver that will solve the equation.Unfortunately, Excel Solver is used manually and it is very timeconsuming. Another way to automatically solve for r_(n) is to use anestimate for r_(n) in the denominator of equation 23. The result of theequation is then a better estimate for r_(n). The new value for r_(n)can then be used in the denominator of equation 23. This process can berepeated until the value for r_(n) converges within desired limits. Agood starting estimate for r_(n) is r_(n-1).

Substituting equation 15 into equation 16:β_(n)=tan⁻¹(r _(n) /z _(n))−tan⁻¹(h/(r _(n) ² +z _(n) ² −h²)^(1/2))  eq. 24z _(n) ≈z _(n-1) −r _(n-1)(Δα+β_(n-1)−β_(n-2))  eq. 19

$\begin{matrix}{r_{n} \cong \frac{t_{n}\;{\cos\left( {{\tan^{- 1}\left( {r_{n - 1}/z_{n - 1}} \right)} - {\tan^{- 1}\left( {h/p_{n - 1}} \right)}} \right)}}{K_{1}\left\lbrack {\left( {p + x} \right) - \left( {r_{n}^{2} + z_{n}^{2} - h^{2}} \right)^{1/2}} \right\rbrack}} & {{eq}.\mspace{14mu} 23}\end{matrix}$

Equation 24 can be substituted into equation 19 to give z_(n) as afunction of z_(n-1), z_(n-2), r_(n-2), Δα, and h.

We finally have all of the equations that we need to solve thedifferential equation for the tangent radius r as a function of pulleyangle a. FIGS. 14 a, 14 b, and 14 c, show the above equations enteredinto a Microsoft Excel spreadsheet. Values for the step angle αΔ, linkangle q, the starting pulley radius r₀, dimensions b, p₀, h, the springconstant K₁, and the torque t are required. The torque can be a constantor an equation. Successively smaller values of the step angle Δα shouldbe tried until the solution converges within desired limits.

As seen in FIG. 13, when the pulley rotates by an angle a_(n), angle βchanges from β₀ to β_(n). As a result, the orientation of the cablerelative to the pulley changes by the cable wrap angle ω, where:ω=α_(n)+β_(n)−β₀  eq. 25

Let's look at an example, assuming the following:

${Constants}\text{:}\mspace{14mu}\begin{matrix}{K_{1} = {14.4\mspace{11mu}{lb}\text{/}{in}}} \\{\theta = {{p/2}\mspace{14mu}{radians}}} \\{b = {3.0\mspace{14mu}{inches}}} \\{h = {{.30}\mspace{14mu}{inches}}}\end{matrix}$${Initial}\mspace{14mu}{Conditions}\text{:}\mspace{14mu}\begin{matrix}{\alpha_{0} = {0{^\circ}}} \\{p_{0} = {12.0\mspace{14mu}{inches}}} \\{r_{0} = {3.0\mspace{14mu}{inches}}} \\{\tau = {{90\mspace{14mu}{in}} - {{lb}.\mspace{14mu}\left( {{constant}\mspace{14mu}{torque}} \right)}}}\end{matrix}$

FIG. 15 shows a graph of the resulting pulley tangent radius r for α=0°to α=200°. Radius r is plotted as a function of the cable wrap angle ω,and not the pulley angle α. When the pulley rotates by α=200°, the cablewraps around the pulley by an angle of ω=191.79°.

FIG. 16 is a drawing of the resulting sliding-pivot spiral pulley. Itwas created by drawing the calculated tangent line for every 5° step ofthe cable wrap angle ω. Line segments were formed between intersectionsof tangent lines. A spline curve was drawn through the midpoints of theline segments. The spline curve represents the centerline or neutralaxis of the cable as it bends around the spiral pulley.

As mentioned earlier, the pulley torque does not need to be a constant.FIG. 17 shows a graph of pulley tangent radius r for a pulley with aparabolic torque profile. All of the other parameters for this pulleyare identical to the previous constant torque pulley. The torque profileis symmetric about α=100°. The torque τ varies from 90 in-lb_(f) atα=0°, up to 180 in-lb_(f) at α=100°, and back down to 90 in-lb_(f) atα=200°.

FIG. 18 shows the new pulley with the parabolic torque profile alongwith the previous constant torque pulley. It can be seen that theparabolic torque pulley is more compact than the constant torque pulley.Not only is the pulley smaller but also it transfers 66.6% more energyover the 200° rotation.

Derivation of the Geometry for a Fixed-Pivot Spiral Pulley

Now let's look at the derivation of the shape of pulley #2 shown inFIGS. 12 a and 12 b. FIG. 19 shows a free body diagram of a spiralpulley with a fixed-pivot. The axis of rotation of the spring pivotpasses through point R. The axis of rotation of the pulley is parallelto the spring pivot axis. The pulley axis passes through point Q. PointsP and Q are separated by a fixed distance k. The centerline of thespring and cable lay in the plane that contains points P and Q and thatis perpendicular to the pulley axis of rotation. The flexible cableextends from the spring to a tangent on the spiral pulley. Dimension ris the tangent radius or perpendicular distance from the pulley axis tothe flexible cable. Angle f is the angle between the flexible cable andthe line through points Q and R.

The spring force is described by the following equation:F ₂ =K ₂ s+f ₀  eq. 26

Where f₀ is the initial tension in the spring and s is the extension ofthe spring.

Angle λ is the angle of rotation of the pulley, starting from theorientation when the spring extension s=0 and F₂=f₀.

The torque τ on the pulley produced by F₂ is:

$\quad\begin{matrix}\begin{matrix}{\tau = {F_{2}r}} \\{= {\left( {{K_{2}s} + f_{0}} \right)r}}\end{matrix} & {{eq}.\mspace{14mu} 27}\end{matrix}$

Solving for r:

$\begin{matrix}{r = \frac{\tau}{\left( {{K_{2}s} + f_{0}} \right)}} & {{eq}.\mspace{14mu} 28}\end{matrix}$

When the pulley rotates by an angle dλ, the resulting spring extensionwill be:ds=rdλ

This can be converted into a finite difference equation.

$\begin{matrix}{{s_{n} - s_{n - 1}} = {r_{n - 1}{\Delta\lambda}}} & {{eq}.\mspace{14mu} 29} \\{s_{n} = {{r_{n - 1}{\Delta\lambda}} + s_{n - 1}}} & \;\end{matrix}$

Now converting equation 28 into a finite difference equation:

$\begin{matrix}{r_{n} = \frac{\tau_{n}}{\left( {{K_{2}s_{n}} + f_{0}} \right)}} & {{eq}.\mspace{14mu} 30}\end{matrix}$

As seen in FIG. 19, when the pulley rotates by an angle λ_(n), angle fchanges from f₀ to f_(n). As a result, the orientation of the cablerelative to the pulley changes by the cable wrap angle ω, where:ω=λ_(n) +f _(n) −f ₀Looking at angle φ: φ=sin⁻¹(r/m)Thus: ω_(n)=λ_(n)+sin⁻¹(r _(n) /m)−sin⁻¹(r ₀ /m)  eq. 31

FIGS. 20 a and b show equations 29, 30, and 31 entered into an Excelspreadsheet.

Let's look at an example, assuming the following:

Constants: K₂ = 14.4 lbf./in f₀ = 30 lbf. m = 12. inches InitialConditions: λ₀ = 0° r₀ = 3.0 inches s₀ = 0.0 inches Torque Profile τ =180-.009(λ-100°)² in-lb. (parabolic torque)

FIG. 21 shows a graph of the resulting pulley tangent radius r for λ=0°to λ=200°. This is a fixed-pivot spiral pulley with a parabolic torqueprofile. It has the same torque profile as the previous sliding-pivotpulley. These two pulleys can be attached to each other tocounterbalance the adjustment of the gravity counterbalance shown inFIG. 12. The two pulleys should be phased relative to each other suchthat sliding-pivot pulley angle α=100° when fixed-pivot pulley angleλ=100°.

5. Link-Angle Compensation and Counterbalance Mechanism

The adjustment counterbalance mechanism shown in FIGS. 12 a and 12 b hasa significant limitation. It balances the adjusting force at only onevalue of (b cos θ). Remember, from equation 7, the force required toadjust the gravity counterbalance at an angle θ is:F ₁ =K ₁(a−b cos θ)

If we assume that the distance b is fixed, then the adjustmentcounterbalance shown in FIGS. 12 a and 12 b will work at only oneabsolute value of link angle ±θ. At another link angle θ=θ₁, the forcerequired from the adjustment counterbalance is:F ₁ =K ₁(a−b cos θ₁)The error is

$\begin{matrix}\begin{matrix}{F_{error} = {{K_{1}\left( {a - {b\mspace{14mu}\cos\mspace{14mu}\theta_{1}}} \right)} - {K_{1}\left( {a - {b\mspace{14mu}\cos\mspace{14mu}\theta}} \right)}}} \\{F_{error} = {K_{1}{b\left( {{\cos\mspace{14mu}\theta} - {\cos\mspace{14mu}\theta_{1}}} \right)}}}\end{matrix} & {{eq}.\mspace{14mu} 32}\end{matrix}$

For a given angle θ₁, the error remains constant, independent ofdimension a. The error force may be positive or negative. The gravitycounterbalance can still be adjusted however.

FIGS. 23 a and 23 b show a modified version of the adjustmentcounterbalance shown in FIGS. 12 a and 12 b. In the modified version,the dual spiral pulley and spring #2 are located on the moving carriagewith spring #1. The force on the carriage is the same for both versions.In FIGS. 12 a and 12 b, the pivot translates relative to the pulley. InFIGS. 23 a and 23 b, the pulley translates relative to the pivot. Therelative motion is the same for both.

In equation 32, if we assume that θ₁=π/2 radians, then F_(error)=K₁ bcos θ

Looking back at equation 8, at θ=θ₁=π/2, the force produced by thecounterbalance mechanism will be:F′ ₁ =−K ₁ p+K ₁(e−b cos θ)=−K ₁ p+K ₁ e

But we know that the error is F_(error)=K₁b cos θ

F′₁ can be corrected by subtracting the known error.

$\quad\begin{matrix}\begin{matrix}{F_{desired} = {F_{1}^{\prime} - F_{error}}} \\{= {{{- K_{1}}p} + {K_{1}e} - \left( {K_{1}b\;\cos\;\theta} \right)}} \\{= {{- {K_{1}\left( {p + {b\;\cos\;\theta}} \right)}} + {K_{1}e}}}\end{matrix} & {{eq}.\mspace{14mu} 39}\end{matrix}$

From equation 33, it can be seen that if θ_(?) π/2, the adjustmentcounterbalance mechanism can be brought back into balance by increasingdimension p by the amount b cos θ.

FIGS. 24 a and 24 b show a modified version of the adjustmentcounterbalance shown in FIGS. 23 a and 23 b. The pivot is mounted on aseparate linear bearing. With the extra linear bearing, dimension p canbe changed by the amount (b cos θ) to compensate for the force error inthe adjustment counterbalance,F_(error)=K₁b cos θ.

FIGS. 24 a and 24 b show a force F₃. This is the force that is requiredto balance the force on the sliding-pivot. Force F₃ has the samenegative stiffness characteristics shown in FIG. 11 b. As a result, asimple extension spring can be used to counterbalance the link anglecompensation mechanism. The spring should have a positive springconstant K₁. It can be referred to as the link angle compensationcounterbalance spring, or spring #3 for short.

The counterbalance force F₃ depends on dimension a. Spring #3 needs tobe properly adjusted or pretensioned. When the gravity counterbalance isadjusted to the desired value of dimension a, force F₃ acting throughthe dual capstan, should balance the vertical component of the force incable #3. The most common value to select for dimension a is thedistance that corresponds to an empty link with zero payload. Choosingthis value of a allows the system to adjust to a new load when it is notholding onto a load. By choosing another value of a, the system caneasily adjust to a new load only while it is already holding onto onespecific payload. The system can be adjusted when it is not balanced,but extra motor power and energy is needed.

Other Versions of the Link-Angle Compensation and CounterbalanceMechanism

FIGS. 25 a and 25 b show a modified version of the counterbalancemechanism of FIGS. 24 a and 24 b. An idler pulley has been attached tothe spring carriage. Both spiral pulleys are now effectively fixed-pivotpulleys. The translating pivot has been replaced with a capstan. Thecapstan allows spring #3 to be folded into a more compact geometry.Another feature of the capstan is that it can have two different radii.

The two radii allow for a fine adjustment of the effective springconstant of spring #3. As before, force F₃ should have a spring constantof K₁. Spring #1 has the constant K₁. It is relatively difficult andexpensive to precisely match spring constants of two or more differentsprings.

The effective spring constant for F₃ can be calculated as follows. Thedirect force from spring #3 acting on cable #4 is F₄. Force F₄ acts onthe capstan at a radius R₁. Force F₃ acts on the capstan at a radius R₂.The sum of the moments on the dual capstan is equal to zero.

Solving for F₃:

$\begin{matrix}{0 = {{F_{3}R_{2}} - {F_{4}R_{1}}}} & \; \\{{Solving}\mspace{14mu}{for}\mspace{14mu} F_{3}\text{:}} & \; \\{F_{3} = {F_{4}{R_{1}/R_{2}}}} & {{eq}.\mspace{14mu} 34}\end{matrix}$

If n is the deflection, and f₄ is the initial tension of spring #3, thedirect force from spring #3 acting on cable #4 can be written:F ₄ =K ₃ n+f ₄

Substituting for F₄ in equation 34:

$\begin{matrix}{{F_{3} = {\left( {{K_{3}n} + f_{4}} \right){R_{1}/R_{2}}}}{\frac{\mathbb{d}F_{3}}{\mathbb{d}n} = {K_{3}\left( {R_{1}/R_{2}} \right)}}{{or}\text{:}}{{\mathbb{d}F_{3}} = {{K_{3}\left( {R_{1}/R_{2}} \right)}{\mathbb{d}n}}}} & {{eq}.\mspace{14mu} 35}\end{matrix}$

The desired force F₃ can be written:F ₃ =K ₁ p+f ₃

If cable #4 is pulled a distance dn, the dual capstan will rotate by anangle do/R₁. If cable #3 moves a distance dp, the dual capstan willrotate by an angle dp/R₂.

Both capstans rotate at the same rate, thus:dp/R ₂ =dn/R ₁.or: dn=(R ₁ /R ₂)dp

Substituting dn into equation 35:

$\begin{matrix}{{{\mathbb{d}F_{3}} = {{K_{3}\left( {R_{1}/R_{2}} \right)}^{2}{\mathbb{d}p}}}{{or}\text{:}}{\frac{\mathbb{d}F_{3}}{\mathbb{d}p} = {K_{3}\left( {R_{1}/R_{2}} \right)}^{2}}{{but}\text{:}}{K_{1} = \frac{\mathbb{d}F_{3}}{\mathbb{d}p}}{{thus}\text{:}}{K_{3} = {K_{1}\left( {R_{2}/R_{1}} \right)}^{2}}} & {{eq}.\mspace{14mu} 36}\end{matrix}$

The ratio R₂/R₁ for the dual capstan can be accurately adjusted to givethe desired value of K₁ from a spring with a given constant K₃.

FIGS. 26 a and 26 b show a modified version of the counterbalancemechanism of FIGS. 25 a and 25 b. The idler pulley on the springcarriage has been eliminated. Cable #3 goes directly from the spiralpulley to the dual capstan. When the spring carriage translates up anddown, spiral pulley #1 acts like a translating pivot pulley. When thespring carriage is fixed and the dual capstan rotates, spiral pulley #1acts like a fixed-pivot pulley.

FIG. 22 is a graph showing the pulley tangent radius for a fixed and asliding-pivot spiral pulley. Both pulleys produce the same torqueprofile with the same spring. The radius curves for both pulleys arevery similar. A spiral pulley can be made using the average of the tworadius curves. Such an “average” profile spiral pulley should work wellfor spiral pulley #1 in FIGS. 26 a and 26 b.

6. Load Compensation and Counterbalance Mechanism

One more adjustment and counterbalance mechanism can be added to thesystem. FIGS. 27 a and 27 b show a system with a load compensation andcounterbalance mechanism. A second dual capstan, along with a second setof spiral pulleys, and another spring has been added to the previoussystem. These components can be referred to as dual capstan #2, dualspiral pulley #2, and spring #4 respectively. Spring #4 can also bereferred to as the load compensation counterbalance spring. Dual capstan#2 can also be referred to as the load compensation adjustment capstan.

From the previous section on link-angle compensation, we know that thepretension on spring #3 affects the gravity counterbalance load level atwhich dual capstan #1 is balanced. Rather than using a fixed preload forspring #3, the preload on spring #3 can be adjusted with dual capstan#2. This allows dual capstan #1 to be adjusted with no work at differentgravity counterbalance load levels.

Spring #4, dual spiral pulley #2, and dual capstan #2 are added tocounterbalance the adjustment of the preload on spring #3. As before, aspring with a negative stiffness is needed to counterbalance a springwith a positive stiffness. Spring #3 has a positive stiffness of K₃.Dual capstan #2, along with dual spiral pulley #2, and spring #4,provide the negative stiffness force.

The stiffness of force F₅ as shown in FIG. 27 can be defined as K₅. Therelationship between the variables K₃, K₅, R₃, and R₄ can be derived inthe same way that equation 36 was derived. This gives us the nextequation.K ₃ =K ₅(R ₃ /R ₄)²  eq. 37

Substituting into equation 36 yields:

$\begin{matrix}{K_{1} = \frac{{K_{5}\left( {R_{1}R_{3}} \right)}^{2}}{\left( {R_{2}R_{4}} \right)^{2}}} & {{eq}.\mspace{14mu} 38}\end{matrix}$

Both of the pulleys in dual spiral pulley #2 act as fixed pivot spiralpulleys. The Excel spreadsheets shown in FIGS. 20 a and 20 b can be usedto derive the shape of the two pulleys. The process is iterative,starting with assumed values for several parameters. These are K₄, m,r₀, the initial force of spring #4 f₄, and the torque profile τ.

7. System Operation

To understand how the counterbalance system operates, it may be usefulto review the advantages and limitations of the various mechanisms. Inorder of increasing complexity, the counterbalance systems fall into thefollowing seven categories:

-   1. Fixed Gravity counterbalance as shown in FIG. 3. The peak    magnitude of the sinusoidal counterbalance torque is fixed. It is    the simplest and the least expensive gravity counterbalance, but it    is also the least flexible. A type #1 counterbalance would be most    appropriate for balancing fixed loads such as lamps, computer    monitors, or hatch covers.-   2. Adjustable gravity counterbalance as shown in FIGS. 9 a and 9 b.    An adjustment mechanism has been added to the type #1 mechanism    above. The magnitude of the counterbalance torque is adjusted by    moving the spring carriage up or down. The gravity counterbalance is    adjustable, but energy is required or lost during adjustment. It is    appropriate for loads that may change, but not often or by a large    amount. The adjustability also enables a more accurate balance,    resulting in lower operating force and smoother motion. An operating    room microscope support arm might be a good application.-   3. Gravity counterbalance with counterbalanced adjustment as shown    in FIGS. 12 a, 12 b, 23 a and 23 b. A mechanism to counterbalance    the adjustment of the gravity counterbalance has been added to the    type #2 mechanism above. The gravity adjustment is balanced at only    one link-angle ±θ′. At other angles, energy is required or lost    during the adjustment. Less energy may be needed than with no    adjustment counterbalance however. The forces on the spring carriage    remain balanced only as long as the link-angle remains at ±θ′. When    the link-angle changes from ±θ′, a force or a locking mechanism or    brake is needed to hold the adjusted position of the spring    carriage. These are appropriate for loads that change often, are    heavy, or where limited power is available. A good application might    be for overhead storage of heavy items. The adjustment should be    balanced in the down position. That way, load can be added or    removed from storage when the storage is accessible. The    counterbalance can then be readjusted with no effort and then    returned to up or any other position. No work is needed to move the    load. A fixed counterbalance would require work to raise or lower a    load. Work is required for the adjustable counterbalance too. The    work isn't done while lifting the load. The work is done during the    adjustment.-   4. Gravity Counterbalance with counterbalanced adjustment and    link-angle compensation as shown in FIGS. 24 a and 24 b. A mechanism    has been added to adjust dimension p independently from dimension a.    This is called the link-angle compensation adjustment. The gravity    counterbalance can now be adjusted at any link-angle θ with no    addition or loss of energy. Unfortunately, energy is still required    or lost during the angle compensation adjustment.-   5. Gravity Counterbalance with counterbalanced adjustment and    link-angle compensation with counterbalance as shown in FIGS. 25 a    and 25 b. A mechanism to counterbalance the link-angle compensation    adjustment has been added to the type #4 mechanism above. At one    fixed position of the spring carriage, dimension a=a′, the net force    on the link-angle compensation adjustment will be zero. At position    a′, the link-angle compensation can be adjusted without any energy.    Note that while the link-angle compensation is being adjusted, the    net force on the spring carriage will remain balanced only if the    link-angle θ changes simultaneously. If the link-angle and the    link-angle compensation are not moved simultaneously, then a force    is required, or the carriage brake should be set to hold the    carriage in place. Unfortunately, the link-angle compensation is    balanced at only one position of the spring carriage. This    corresponds to only one gravity counterbalance load. When the spring    carriage is moved to counterbalance a different load, the link-angle    compensation adjustment becomes unbalanced. At this time, the    compensation adjustment should be held or the brake should be set on    the adjustment. Once the link-angle compensation brake is set, the    gravity counterbalance adjustment will be balanced at only one    link-angle again. The link must be returned to that angle before the    gravity counterbalance can be adjusted without any energy. Sequence    of operation is important. Angle compensation should be adjusted    first so that the net force on the spring carriage is zero. Then the    angle compensation adjustment should be locked. Then the spring    carriage can then be adjusted to a new desired load (dimension a).    Finally, the carriage should be locked so that it won't move due to    the resulting force imbalance. The link-angle compensation    adjustment must be locked before the load is adjusted. As a result,    angle compensation only works at one load level. Angle θ″ must be    maintained while the load is adjusted. The load adjustment can then    be locked, and the link-angle θ can be rotated to any θ. The link    must be returned to the specific θ″ before the carriage can be    unlocked and adjusted again. Only when the carriage is adjusted back    to W₁ can the link-angle compensation be reengaged without loss.-   6. Gravity counterbalance with counterbalanced adjustment,    link-angle compensation with counterbalance, and load compensation,    as shown in FIGS. 26 a, 26 b, 27 a and 27 b. A mechanism has been    added to adjust the preload on spring #3. This is called the load    compensation adjustment. The preload on spring #3 affects the    position of the spring carriage or the load level at which the    link-angle compensation adjustment is balanced. The load    compensation adjustment allows the link-angle compensation    adjustment to be made without any energy. Unfortunately, energy is    still required or lost during the load compensation adjustment. At    one position of the spring carriage, dimension a =a′, the net force    on the link-angle compensation adjustment will be zero. At position    a′, the link-angle compensation can be adjusted without any energy.-   7. Gravity counterbalance with counterbalanced adjustment,    link-angle compensation with counterbalance, and load compensation    with counterbalance as shown in FIGS. 28 a and 28 b. A mechanism to    counterbalance the load compensation adjustment has been added to    the type #6 mechanism above. This system is the most flexible. The    counterbalance can be adjusted at any angle and then readjusted at    another angle to a different load level. This system is most useful    for robotic applications in which a payload may be picked up or    dropped off at different elevations.    8. Multiple Counterbalance Mechanisms on the Same Axis of Rotation    Dual Opposed Counterbalance

Multiple gravity counterbalance mechanisms can be connected to the sameaxis of rotation. FIGS. 30 a and 30 b show two counterbalance mechanismsthat are arranged on opposite sides of the axis of rotation. Eachmechanism can be adjusted by moving its spring carriage along a verticalpath. The torque from the lower mechanism about the axis of rotation is:T ₁ =−a ₁ bK ₁ sin θ

The torque from the upper mechanism is:T₂=a₂bK₂ sin θ

The sum of the torques is:T ₁ +T ₂ =b(a ₂ K ₂ −a ₁ K ₁) sin θ

This torque will balance a load MgL sin θ when:MgL=b(a ₂ K ₂ −a ₁ K ₁)  eq. 39

One advantage of the dual opposed counterbalance mechanism is that itallows small loads to be balanced throughout the full 360° rotation ofthe axis. With only one counterbalance mechanism, to balance a smallload, dimension a must be small. But when dimension a is small, thepivot bearing on axis-A interferes with the idler pulley on the cablegimbal. From equation 39, the dual opposed counterbalance mechanism canbe adjusted to balance a zero load by adjusting a₁ and a₂ so that:a ₂ K ₂ −a ₁ K ₁=0

Both minimum values of dimensions a₁ and a₂ can be large enough to avoidinterference between the pivot bearing and the idler pulleys. Note thatspring constants K₁ and K₂ do not need to be equal to each other. Thedual counterbalance mechanism can be adjusted by moving either one ofthe spring carriages, or it can also be adjusted by moving both of thespring carriages. The system can be simplified by fixing the uppercarriage and adjusting only the lower carriage. This eliminates the needfor linear bearings on the upper carriage. The two counterbalancemechanisms shown in FIGS. 30 a and 30 b are type #2 adjustablemechanisms described earlier. Any of the other types of counterbalancemechanisms can be substituted. For example, if the upper mechanism is asimple fixed type #1 mechanism, and the lower mechanism is a completetype #7 mechanism, the resulting dual mechanism has all of theadjustability of the type #7 mechanism.

Multiple Mechanisms for Adjustable Phase and Magnitude

FIGS. 31 a and 31 b show a system with four counterbalance mechanismsacting on one axis of rotation. The mechanisms are oriented at 90°intervals around the axis. Each one of the mechanisms delivers a torquethat varies sinusoidally with the link angle θ. Each of the sinusoids isphased 90° apart. If two or more adjacent mechanisms are adjustable,it's possible to deliver a sinusoidal torque with an arbitrary phase andmagnitude.

The ability to adjust the phase of the torque has advantages. Forexample, a system may become tilted relative to gravity. The phaseadjustment allows the system to compensate for the tilt. With theappropriate type of individual mechanisms, the phase can be adjustedwithout consuming any energy.

Up until now, we have assumed that the counterbalance system would beused to balance the gravity torque on a link. A counterbalance mechanismcan be used to deliver a torque that is not a function of gravity alone.For example, if the link is part of a robot arm, the arm will applyforces in various directions, not just up or down. The ability to adjustthe phase of the counterbalance torque allows a link to deliver a forcein any direction. The torque may even be used to accelerate ordecelerate the link. This can all be done with negligible energyconsumption.

Dual Phase Shifted Counterbalance Mechanism

FIGS. 32 a and 32 b show a system with two cable gimbal mechanisms. Eachof the cables is connected to the same pivot bearing on axis-A. Eachcable gimbal mechanism is mounted on a pivot that rotates about axis-C.This is the same axis that the link rotates about. The distance a is aconstant for both mechanisms. Angle b is the angle from the horizontalto the line CB for each mechanism.

The magnitude of the counterbalance torque will be zero when angle β=0°.When angle β=90°, both counterbalance torques add together. The nettorque is:T _(net)=−2abK ₁ sin θ sin β  eq. 40

This system can be adjusted to zero torque without any mechanicalinterference. Unlike the dual opposed counterbalance mechanism, bothspring mechanisms act together to produce a higher peak torque with thesame springs. The phase of the net torque can be adjusted by rotatingthe individual cable gimbal mechanisms to different angles.

9. Translational Counterbalance Mechanisms

The previous gravity counterbalance mechanisms were designed for linkswith rotary joints. They provide a torque to balance the gravity momentat the joint. Prismatic or translational joints can be counterbalancedtoo. A constant force mechanism is required to counterbalance atranslating link.

FIGS. 35 a and 35 b show a constant force mechanism. The mechanism issimilar to the rotary counterbalance shown in FIGS. 23 a and 23 b. Therotating link and the cable gimbal have been eliminated.

In FIGS. 35 a and 35 b, the translating link is called the springcarriage. The carriage is constrained by a linear bearing so that it hasonly one translational degree of freedom and zero rotational degrees offreedom. The path of the bearing is oriented at an angle ψ relative tovertical. The force on the carriage, in the direction of travel, fromspring #1 is F₁, where:F ₁ =K ₁ a+f ₁  eq. 41

The dual spiral pulley, in combination with spring #2 produces a forceon the carriage in the direction of travel equal to F₃, where:F ₃ =K ₁ p+f ₃

The spiral pulley and spring mechanism can be designed using the methodspreviously discussed. As before, it can be designed with a negativestiffness equal in magnitude to the stiffness of spring #1.K ₃ =−K ₁

Substituting the above equations:F ₃ =−−K ₁ p+f ₃  eq. 42

If M_(c) is the mass of the carriage, g is the gravitationalacceleration, and “a” is the acceleration of the carriage, summing allforces on the link in the direction of travel:

$\begin{matrix}{{0 = {F_{1} - F_{3} - {M_{c}g\mspace{11mu}\cos\;\Psi} - {M_{c}a}}}{{M_{c}\left( {{g\mspace{11mu}{\cos\Psi}} + a} \right)} = {F_{1} - F_{3}}}} & {{eq}.\mspace{14mu} 43}\end{matrix}$

If we assume for now that the carriage is not accelerating, then theforce required to counterbalance the carriage is:M _(c) g cos ψ=K ₁(a+p)+f ₁ −f ₃  eq. 44

For a given M_(c), g, and ψ, the counterbalance force M_(c)g cos ψ isconstant and independent of the position of the link along the linearbearing. If the load

M_(c)g cos ψ changes, the counterbalance force must be adjusted torebalance the link.

FIGS. 33 a, 33 b, and 33 c show how the forces F₁ and F₃ change as thecarriage moves from one extreme position where a=0 to the oppositeextreme where p=0. For fixed values of spring preload force f₁ and forcef₃, the difference between force F₁ and F₃ is constant. As a result, thecounterbalance force (F₁−F₃) is constant. Note that (a+p) also remainsconstant as the carriage moves from one end to the other.

In FIG. 33 a, the spring preload forces, f₁ and f₃, result in a negativecounterbalance force (F₁−F₃).

In FIG. 33 b, the preload force f₁ of spring #1 has been increased sothat the counterbalance force (F₁−F₃)=0. In figure 35, the increase inforce f₁ would be accomplished by tightening the spring #1 forceadjustment nut. This pulls on cable #1 and increases the spring force.

In FIG. 33 c, the preload force f₃ of the dual spiral pulley and springmechanism has been decreased so that the counterbalance force (F₁−F₃) isnow positive. In FIG. 35, the decrease in force f₃ would be accomplishedby lengthening dimension p without moving the spring carriage.

The counterbalance force can be adjusted from a positive to a negativeforce by adjusting the preload force of either spring mechanism or byadjusting both.

FIGS. 36 a and 36 b show a modified version of the translationalcounterbalance in FIGS. 35 a and 35 b. Both of the springs and the dualspiral pulley have been removed from the carriage. The resultingcounterbalance system has less inertia. The smaller carriage sweeps outless volume as it moves from one end of travel to the other.

FIGS. 37 a and 37 b show a modified version of the translationalcounterbalance in FIGS. 36 a and 36 b. A dual capstan has been added.This allows the stiffness of spring #1 to be accurately adjusted so thatit matches the stiffness of the dual spiral pulley mechanism.

In FIGS. 38 a and 38 b, a third spring along with a dual capstan andbrake have been added to the counterbalance. The extra spring acts tocounterbalance the adjustment of the counterbalance force. Theadjustment is balanced at only one position of the spring carriage. Atother positions, the brake on the dual capstan should be set.

FIGS. 39 a and 39 b show a modified version of the translationalcounterbalance in FIG. 38. A motor has been added to automate the forceadjustment.

In FIGS. 40 a and 40 b, a fourth spring along with another dual spiralpulley and dual capstan and brake have been added to the counterbalance.The additional components allow the counterbalance force to be adjustedat any position of the spring carriage.

The counterbalance systems in FIGS. 38 a, 38 b, 39 a, 39 b, 40 a, and 40b all have counterbalanced adjustment mechanisms. In each case, theadjustment counterbalance spring #3 is connected to the negativestiffness spring #2 assembly. As shown earlier, the counterbalance forcecan be adjusted by changing the preload force on either spring #1 orspring #2. The adjustment of spring #1 can be counterbalanced too.Spring #1 is a common positive stiffness spring. As a result, a negativestiffness spring assembly is needed to counterbalance the adjustment ofspring #1.

10. A Rotary Counterbalance Made from a Scotch Yoke and TranslationalCounterbalance Mechanism

FIGS. 41 a and 41 b show a rotary gravity counterbalance mechanism. Thetranslational counterbalance system shown in FIGS. 36 a and 36 b iscombined with a Scotch Yoke mechanism to counterbalance the gravitymoment of a rotary link.

Equation 44 shows the criteria for counterbalancing the translatingcarriage. The carriage by itself will be balanced when:0=M _(c) g cos ψ−K ₁(a+p)+f ₃ −f ₁

The net force on the Scotch Yoke mechanism is F_(Y), where:F _(Y) =M _(c) g cos ψ−K ₁(a+p)+f ₃ −f ₁

The torque produced by the Scotch Yoke mechanism is:

$\begin{matrix}{{T_{y} = {F_{y}b\;\sin\;\theta}}{T_{y} = {\left\lbrack {{M_{c}g\mspace{11mu}{\cos\Psi}} - {K_{1}\left( {a + p} \right)} + f_{3} - f_{1}} \right\rbrack b\;\sin\;\theta}}} & {{eq}.\mspace{14mu} 45}\end{matrix}$

From equation 1, the gravity torque produced by the rotary link is:T₁=MgL sin θ

The rotary link will be balanced when:

$\begin{matrix}{0 = {T_{1} + T_{Y}}} & {{eq}.\mspace{14mu} 46} \\{{{MgL}\mspace{14mu}\sin\mspace{14mu}\theta} = {\left\lbrack {{{Mcg}\mspace{14mu}\cos\mspace{14mu}\psi} - {K_{1}\left( {a + p} \right)} + f_{3} - f_{1}} \right\rbrack b\;\sin\mspace{14mu}\theta}} & \; \\{{MgL} = {\left\lbrack {{{Mcg}\mspace{14mu}\cos\mspace{14mu}\psi} - {K_{1}\left( {a + p} \right)} + f_{3} - f_{1}} \right\rbrack b}} & \;\end{matrix}$

The spring preload forces, f₁ and f₃, can be adjusted to bring therotary link into balance.

Any of the different translational counterbalances can be used in placeof the one shown in FIGS. 41 a and 41 b.

Multiple Scotch Yoke counterbalance mechanisms can be connected to thesame axis of rotation. From equation 45, it can be seen that the sinefunction can be multiplied by a positive number, a negative number, orzero. Two of the Scotch Yoke counterbalance mechanisms, phased 90°apart, can be connected to the same axis of rotation. With thisarrangement, it's possible to deliver a net sinusoidal torque with anyphase and magnitude.

11. Extending the Counterbalance to Multiple Degrees of Freedom

The zero-length spring rotary counterbalance system was analyzed in thefirst section. Until now, the counterbalanced link has been shown withone degree of freedom. The counterbalance is not limited to one degreeof freedom.

Let's look back at the criteria for the zero-length springcounterbalance. First, the link should be constrained in all threetranslational degrees of freedom at point C. Second, the springmechanism should deliver a force F that acts along the line thatintersects point A on the link and point B on the cable gimbal axes.

The key to counterbalancing more than one rotational degree of freedomlies in the mechanical constraints at points A and B. As the linkrotates, the spring mechanism attachment at point A should not introducea moment to the link, and the spring mechanism attachment at point Bshould not introduce a moment to the spring mechanism. In other words,the degree of rotational freedom at the attachment points must be largeenough to avoid any rotational constraint.

In all of the previous systems, the link was constrained at point C sothat it was free to rotate about only one axis. As a result, only onerotational degree of freedom was needed at the cable gimbal and only onerotational degree of freedom was needed at point A.

There is a variety of options for counterbalancing the link as itrotates about more than one axis. At the point A cable attachment, aball joint can be used.

A ball joint can provide a complete three rotational degrees of freedom.Ball joints typically have a limited range of motion however.

A universal joint can also be used at the cable attachment point. FIGS.42 a and 42 b show a conventional u-joint consisting of two yokes with aspider between them. A pitch axis passes through yoke #1 and the spider.A yaw axis passes through yoke #2 and the spider. The pitch and yaw axesintersect at point A of the link. Yoke #1 is attached to the link, andyoke #2 is attached to the cable. Two roll axes are also shown. Rollaxis #1 allows for rotation between yoke #1 and the link. Roll axis #2allows for rotation between yoke #2 and the cable. Both roll axes alsointersect at point A.

Not all of the four u-joint axes shown are needed for every application.Remember that in all of the previous systems there was only one axis ordegree of freedom at point A. The number of axes needed at point Adepends on the rotational freedom at the point C link pivot and theorientation of the axes at both points A and C. For example, if the axisof rotation of the link at point C is vertical, then the angle betweenthe cable and the link does not change. In this case, zero axes areneeded at point A. The link might be part of a boom and thecounterbalance system would be used to eliminate the moment from thepivot bearing at point C.

Depending on the application, the u-joint in FIGS. 42 a and 42 b may beeliminated, or it can be used with one, two, three, or four of its axes.The redundant fourth degree of freedom may be needed for a large rangeof motion.

When the u-joint pitches or yaws by 90°, it is at a singularity. It nolonger has 4 degrees of freedom. At 90° pitch or yaw, two of the u-jointaxes are aligned with each other and there are only 3 degrees offreedom.

Several two degree of freedom cable gimbal mechanisms were discussed insection 2. Any one of the cable gimbal mechanisms can be combined with au-joint at point A. If the u-joint at point A allows for rotation aboutthe cable axis, then a third degree of freedom for the cable gimbal isnot needed. For example, a link can be counterbalanced about any axis ofrotation passing through point C by combining any of the two degree offreedom cable gimbals with a four-degree of freedom universal joint atpoint A.

There are several approximate methods of providing rotational freedom atpoints A and B. Depending on the cost, accuracy, range of motion andother system requirements, the approximate systems may be preferable.For example, the cable can be fixed at point A with no pivot mechanism.The bending and torsional stiffness of the cable will introduce a momenterror and a force direction error at point A.

In some situations, a single idler pulley can be substituted for thecable gimbal mechanism. With this simplification, the apparent positionof point B will not be fixed and there will be an error in the distanceC between points A and B.

With the zero-length spring counterbalance, the link can becounterbalanced about all axes passing through point C even if there isno freedom for the link to rotate about the axis. The next section showsa variety of arrangements for the joint axes. Most systems are shownwith the adjacent axes orthogonal to each other. The yaw axes are oftenshown vertical and the pitch axes are shown horizontal. With thezero-length spring counterbalance, the adjacent axes do not need to beorthogonal, and all axes may have any orientation relative to gravity.

If a moment is introduced by the spring mechanism at point A, then theconstraint on the link at point C must support this moment.

12. Extending the Counterbalance to Multiple Link Arms

The previous systems were designed for counterbalancing a single rigidbody or link. Arms with two or more links in series can becounterbalanced too.

Pantograph Mechanisms

The analysis of the zero-length spring counterbalance mechanism assumedthat the force from the spring acted directly on the link at point A. Itassumed that point A was located on the line that passes through point Cand the center of gravity of the link. It assumed that point B waslocated on the line that passes vertically through point C.

With the above constraints, it's difficult to spring counterbalance amultiple link arm. For example, assume a two-link arm with an upper armlink and a forearm link. The upper arm is connected to ground by ashoulder joint, and the forearm is connected to the upper arm by anelbow joint. A vertical gravity reference is needed to counterbalanceeach link. The upper arm is next to ground for its vertical reference.The forearm usually does not usually have a vertical reference on theadjoining upper arm link. The one exception is when the shoulder jointhas only a vertical axis of rotation.

The above limitations can be avoided with the following mechanism.Another link can be added at a location remote from the link to bebalanced. The two links can be mechanically coupled so that the remotelink copies the relevant angular motion of the link to be balanced. Ifthe remote link is at a location with a vertical reference, then aspring counterbalance can be connected to the remote link. The springcounterbalance will balance both links. The mechanism that couples thelink to the remote link is a pantograph mechanism. We can call theremote link the pantograph link.

Reasons for Using a Pantograph Mechanism

The pantograph link does not need to have the same length, mass, orinertia of the link that is being balanced. In the case of the abovetwo-link serial arm, the pantograph link for the forearm can be locatedat the shoulder joint at a location with a vertical reference. Thisarrangement provides several benefits. First, it provides the verticalreference for the forearm. Second, it moves the mass of thecounterbalance system closer to the shoulder rotation axes. Therotational inertia of the arm about its shoulder axes will be less.Third, if the arm has joint motors, the elbow motors can be located inthe shoulder and connected to the pantograph too. This will decrease thearm inertia. Finally, if there isn't space available at a joint, thepantograph can be used to move the counterbalance to a differentlocation.

Examples of Pantograph Mechanisms

Axial Offset Pantograph

Many of the figures have shown a simple pantograph mechanism. In FIGS. 4a and 4 b for example, the link is mounted on a pair of bearings. Thebearings limit the link rotation to only one axis. We can call this axisC. The counterbalance system should support the gravitational momentabout axis C. Other moments are supported by the two bearings. Point Cin FIG. 1 has been replaced by axis C.

The counterbalance torque is produced by the zero-length springmechanism and the pantograph link. The torque is coupled to the link bythe shaft that connects the two links. In this case, the shaft can bethought of as a very simple pantograph mechanism. The angle at one endof the shaft is reproduced at the other end of the shaft. The shaft hasbeen used to axially offset the counterbalance mechanism to a newlocation.

Many other mechanisms can be used to axially offset the link from thecounterbalance mechanism. One or more universal joints can be used inseries on the shaft. Some u-joints do not have “constant velocity” or a1 to 1 input/output ratio. They can usually be used in pairs to producea 1 to 1 ratio. Flexible shafts or flexible u-joints may also be used. Aparallel link Schmidt type coupling may also be used. The Schmidtcoupling allows both axial and radial offset of the counterbalancemechanism.

Phase Shifting Pantograph

FIG. 43 shows another very simple form of pantograph mechanism. Insection 1, φ was defined as the angle between the vertical and the linefrom point C to the link center of gravity. In the phase shiftingpantograph mechanism, angle φ has been copied at an orientation offsetfrom vertical by phase angle f. The zero-length spring is oriented atthe offset phase angle f.

It should be noted that this very simple pantograph mechanism works foronly one axis. If the link is free to rotate about an axis other thanthe phase shift axis, then the moments will not balance properly. If thelink is connected by a joint with more than one degree of freedom, thena more complex pantograph mechanism is required. The pantographmechanism must duplicate the angular motion of the link.

One or Two Degree of Freedom Pantograph

FIG. 44 shows a two degree of freedom pantograph mechanism. The angularmotion of the forearm is duplicated by the pantograph link. The elbowpitch angle is copied at the pantograph pitch axis. The elbow roll angleis copied at the pantograph roll axis. The forearm extends in onedirection from the elbow pitch axis. In FIG. 44, a “phantom” extensionof the forearm is drawn in the opposite direction. Point A is shown aspart of this “phantom” extension. The position of point A moves as if itwas part of the same rigid body as the forearm link. Point A′ is locatedon the pantograph link. The forearm can be balanced by connecting thezero-length spring at point A′.

In the FIG. 44 mechanism, the elbow roll axis is coincident with thepantograph roll axis. As a result, roll axis motion and torque at theelbow joint is transmitted by the upper arm, directly to the pantographjoint.

Pitch axis motion is a little more complex. The forearm link is rigidlyconnected to a capstan or drum. The pantograph link is rigidly connectedto a drum with the same diameter. The forearm link and drum aresupported and constrained by bearings to rotate about the elbow pitchaxis. The pantograph link and drum are similarly supported andconstrained about the pantograph axis. One or more flexible cablesconnect the two drums so that they rotate in the same direction at aratio of 1 to 1. For torque transmission, the cable ends can be rigidlyfastened to the drums. Alternatively, the cables can be tensioned sothat friction will enable torque to be transmitted from one drum to theother. FIG. 44 also shows four optional idler pulleys. These pulleyspinch the cables together so that they will pass through an upper armwith a smaller diameter.

FIG. 44 shows a yaw axis at both joints. The pantograph does not haveany yaw freedom of rotation. That does not mean that the forearm needsto be perpendicular to the pitch axis. The forearm and pantograph linkcan have fixed matching yaw angles. The yaw angle of the link is theangle to the center of gravity of the link, not to the outside geometryof the link. The elbow and pantograph pitch axes should be parallel toeach other. The pitch axes should intersect the common roll axis. Thetwo pitch axes do not need to be perpendicular to the roll axis.

The two degree of freedom pantograph can be converted to a one degree offreedom pantograph by locking one of the two rotational freedoms. Forexample, the upper arm must be mounted on bearings to have roll axisfreedom. If the upper arm is rigidly mounted, then the pantograph willhave only pitch axis freedom.

Three Degree of Freedom Pantograph

FIGS. 45 a, 45 b, and 45 c show a three degree of freedom pantographmechanism. The angular motion of the forearm is duplicated in all threedegrees of freedom by the pantograph link. The elbow and the pantographjoints are mirror images of each other. Each joint is a universal joint,with its inner yoke rigidly connected to the upper arm link. Each jointhas an outer yoke. The outer yoke of the elbow joint is rigidly attachedto the forearm link. The outer yoke of the pantograph joint is rigidlyattached to the pantograph link. Each joint has a spider that is locatedbetween its pair of yokes. The pitch axis passes through the outer yokeand spider of each joint. The yaw axis passes through the inner yoke andspider of each joint. At each joint, the roll, yaw, and pitch axesintersect at a point. At each joint, the adjacent axes do not need to beperpendicular to each other. For example, the elbow yaw axis does notneed to be perpendicular to the elbow roll axis. The correspondingangles on the elbow and pantograph joints need to be the same. The pitchaxes need to be parallel to each other.

The roll motion and torque is transmitted by the upper arm as it was inthe FIG. 44 pantograph mechanism. Yaw axis motion is transmitted in thefollowing way. The spider of each joint is rigidly coupled to a yawcapstan. One or more flexible cables connect the two capstans so thatthey rotate in the same direction at a ratio of 1 to 1. Pitch axismotion is more complex. A bevel gear is rigidly attached to the outeryoke of each joint. They are called the outer pitch gears. Another bevelgear is mounted on bearings that rotate about the yaw axis of eachjoint. These are called the inner pitch gears. On each joint, the outerand inner pitch gears mesh with each other. Each inner pitch gear isrigidly coupled to a pitch capstan. One or more flexible cables connectthe two pitch capstans so that they rotate in the same direction at aratio of 1 to 1. Idler pulleys can be added as in the FIG. 44 pantographmechanism.

Other Parallel Axis Pantograph Mechanisms

In FIGS. 44 and 45 a-45 c, motion and torque is transmitted from oneparallel axis to the other. There are many other ways to accomplishthis. The flexible cables and capstans can be replaced by flat belts andpulleys, tooth belts and pulleys, metal bands and pulleys, or chain andsprockets. Two pairs of bevel gears and a shaft can also transmit themotion. For example, the shaft can be parallel to the upper arm link.One pair of bevel gears would be at each end of the shaft to change theaxis of rotation by 90°. A series of gears can also transmit the motion.An odd number of gears will produce rotation in the same direction ateach end. Over a limited range of motion, a parallelogram four-bar linkwill transmit motion between parallel shafts. The four-bar link has fourlinks in series, with the end of the last link connecting to the startof the first link. In a parallelogram four-bar mechanism, opposite linkshave the same length. To avoid a singularity, a single four-barmechanism is limited to less than + or −90° rotation. Two parallelogramfour-bar mechanisms phased 90° apart can be used to couple two parallelaxes. The two four-bar mechanisms can deliver unlimited rotation. Thisis the same arrangement that steam engines used. The cranks on each sideare shifted by 90°.

Pantograph Mechanisms in Series

More than one pantograph mechanism can be used in series to transmitmotion from the link to the counterbalance mechanism. The number ofdegrees of freedom is limited to the individual pantograph with thelowest number. If the last pantograph mechanism in the series has avertical reference, it can be connected to the counterbalance mechanismat that location. A series of pantograph mechanisms may be needed toreach a vertical reference. For example on a three serial link arm, forthe outer joint, the series of pantograph mechanisms may need to passthrough the next two inner joints to get back to a vertical reference.

An alternate approach may be used for counterbalancing a joint. Apantograph mechanism can be used to transfer a vertical reference out tothe joint. The spring counterbalance mechanism can then be located atthe outer joint.

13. Examples of Counterbalanced Two Link Arms

FIGS. 47 a through 54 c show a variety of two link arms with differentdegrees of freedom and ranges of motion. Most of them are shown withfixed spring counterbalance mechanisms. Any of these can be converted tofully adjustable counterbalance mechanisms. In all of the figures, thejoint axes are labeled and numbered. The numbering starts from groundand works out to the end of the arm. In all of the arms, axis number oneis a vertical axis at the shoulder joint. Vertical axes are inherentlybalanced relative to gravity. As a result, the details of the verticalhave been omitted.

FIGS. 47 a, 47 b and 47 c show a three degree of freedom arm. Theshoulder joint has yaw and pitch axes. The elbow joint has a pitch axis.The upper arm, forearm, and pantograph assembly are the same as shown inFIG. 44. Axis #2, the shoulder pitch axis, is counterbalanced by anaxial offset pantograph mechanism combined with a zero-length spring andgimbal mechanism. The axial offset pantograph mechanism is used to movethe shoulder counterbalance mechanism, to make room for the elbowcounterbalance mechanism.

The arm in FIG. 44 has been shown with the shoulder and elbow pitch axesin a horizontal orientation. Because of this, only one rotational degreeof freedom is needed at each counterbalance cable attachment. The cablegimbals need only one degree of freedom. The shoulder and elbow pitchaxes do not need to be horizontal. If a pitch axis is not horizontal,then the cable gimbal must have at least two degrees of freedom. Thecable attachment should be converted to a u-joint with three or moredegrees of freedom. The shoulder and elbow pitch axes do not need to beparallel.

Mounting Constraints for the Pantograph Axis

The elbow pantograph axis should be aligned so that it intersects theshoulder pitch axis. If it does not, then both the elbow and shouldercounterbalances will not work properly. There will be a coupling betweenthe shoulder joint and the pantograph joint.

In general, to avoid coupling between the joint counterbalancemechanisms, the pantograph axis should intersect all of the non-verticalaxes of the local arm joint. With this alignment, the elbowcounterbalance mechanism in FIGS. 47 a, 47 b and 47 c will balance theforearm moment at the elbow joint and the moment from the elbow jointwill not couple into the shoulder pitch axis.

The forearm still has an affect on the moment at the shoulder pitchaxis. The weight of the forearm acts to increase the moment at theshoulder pitch axis. The weight of the forearm acts as if it wasconcentrated along the elbow axis. This concentrated weight needs to beadded to calculate the moments of the upper arm about the shoulder axes.An effective center of gravity and weight can be calculated for theupper arm in combination with the forearm. For example, in FIGS. 47 a,47 b and 47 c, the mass of the forearm link maps onto the elbow pitchaxis at a point. This point is at the intersection of the joint axiswith the plane that is both perpendicular to the axis and that containsthe center of gravity of the forearm link. The pantograph andcounterbalance mechanisms effectively resolve the load at each jointinto a force and a moment.

For an elbow joint with more than one axis, the axes should intersect.The weight of the forearm link will act at the intersection point.

This process can be repeated for arms with more than two links inseries. At each joint, a pantograph mechanism can be used to support themoment load. The weight of all of the distal links acts on the axis ofrotation of the joint. To avoid coupling the moments into the proximaljoints, a pantograph joint or a new pantograph mechanism can be used ateach proximal joint. Each pantograph axis should intersect the axis ofthe proximal joint so that a moment isn't coupled into the proximaljoint. The counterbalanced moment is transmitted back to ground withoutaffecting the joints in between.

FIGS. 48 a, 48 b and 48 c show a four degree of freedom arm. Theshoulder joint has yaw and pitch axes. The elbow joint has roll andpitch axes. This difference between this arm and the last one is that aset of bearings has been added. The bearings allow the elbow to rotateabout a roll axis. With the extra degree of freedom, a three or fourdegree of freedom u-joint is needed at the cable connection to the elbowpantograph link. A two degree of freedom cable gimbal is needed for theelbow joint counterbalance. The pantograph roll and pitch axes shouldintersect the shoulder pitch axis at one point.

FIG. 50 shows another four degree of freedom arm. The shoulder joint hasyaw and pitch axes. The elbow joint has pitch and yaw axes. The upperarm, forearm, and pantograph assembly are the same as shown in FIGS. 45a, 45 b and 45 c. A three or four degree of freedom u-joint is needed atthe cable connection to the elbow pantograph link. A two degree offreedom cable gimbal is needed for the elbow joint counterbalance. Thepantograph pitch and yaw axes should intersect the shoulder pitch axisat one point.

FIG. 51 shows a five degree of freedom arm. The shoulder joint has yawand pitch axes. The elbow joint has roll, pitch, and yaw axes. Thisdifference between this arm and the last one is that a set of bearingshas been added. The bearings allow the elbow to rotate about a rollaxis. A three or four degree of freedom u-joint is needed at the cableconnection to the elbow pantograph link. A two degree of freedom cablegimbal is needed for the elbow joint counterbalance. All of thepantograph axes should intersect the shoulder pitch axis at one point.

FIGS. 49 a, 49 b and 49 c show a three degree of freedom arm. All jointaxes on this arm can have an unlimited range of motion. The shoulderjoint has yaw and pitch axes. The elbow joint has a pitch axis. Theupper arm, forearm, and pantograph assembly are similar to the mechanismshown in FIG. 44. In FIGS. 49 a, 49 b and 49 c, the upper arm and theforearm have been offset. The offset allows both the upper arm and theforearm to rotate without interference. The shoulder pitch axis has anaxial offset pantograph. The shoulder pantograph is connected to anotherpantograph consisting of two capstans with cables. The far capstanconnects to the shoulder pantograph link. The link connects to thecounterbalance spring mechanism. The elbow axis is pantographed severaltimes, first by an axial offset at the elbow, then by the mechanism ofFIG. 44, then by another axial offset to the rear of the arm. At therear, the elbow pantograph link connects to the spring mechanism.

FIGS. 52 a, 52 b and 52 c show shows an arm with five degrees offreedom. The shoulder joint has yaw, roll, and pitch axes. The elbowjoint has roll and pitch axes. The elbow counterbalance is very similarto the elbow counterbalance in FIGS. 48 a, 48 b and 49 c. The elbowspring and cable have been rerouted. Another degree of freedom has beenadded to the shoulder joint. This is shoulder roll axis #2.

To counterbalance the shoulder joint, another pantograph consisting oftwo capstans and cables has been added. This can be seen in the righthand view. The shoulder pantograph link connects to the far capstanthrough a shaft.

FIGS. 53 a, 53 b and 53 c show an arm with six degrees of freedom. Theshoulder joint has yaw, roll, yaw, and pitch axes. The elbow joint hasroll and pitch axes. The elbow counterbalance is very similar to theelbow counterbalance in the previous

The shoulder has one more degree of freedom than the previous arm did.It's possible but cumbersome to “pantograph” around the additional axis.Another approach has been taken to counterbalance the shoulder. A springand cable gimbal mechanism is connected directly to the upper arm. Thecable gimbal is above the shoulder joint. As a result, the cableconnects to the “front” end of the upper arm link. This arrangementkeeps the shoulder counterbalance from interfering with the elbowcounterbalance.

The shoulder counterbalance cable connects to the arm through a threedegree of freedom joint. The joint has a yoke that spans the upper arm.A bearing surrounds the upper arm. The bearing axis is coaxial with theelbow roll axis #5. The yoke is attached to the outer race of thebearing through trunnion pivot bearings. The cable connects to the yokethrough a bearing. The axis of rotation of the bearing is coincidentwith the cable centerline. All three of the axes intersect at one point.This point should be point A for the upper arm link.

FIGS. 54 a, 54 b and 54 c show a detailed drawing of a four degree offreedom arm. The shoulder joint has yaw and pitch axes. The elbow jointhas roll and pitch axes. This arm has the same kinematics as the armshown in FIGS. 48 a, 48 b and 49 c.

In FIGS. 54 a, 54 b and 54 c, the fixed counterbalance mechanisms havebeen replaced with adjustable counterbalance mechanisms similar to theone shown in FIGS. 27 a and 27 b. The gravity counterbalance mechanismshave counterbalanced adjustment, link-angle compensation withcounterbalance, and load compensation. An additional idler pulley hasbeen added to each counterbalance. The pulleys enable spring #1 andspring #2 to be aligned parallel to the load adjustment direction. As aresult, the counterbalance mechanisms use less volume.

In FIGS. 54 a, 54 b and 54 c, the components of the shoulder jointcounterbalance are labeled, and the elbow joint components are notlabeled. The elbow counterbalance has a two degree of freedom cablegimbal. The shoulder counterbalance has a one degree of freedom cablegimbal. Except for this one difference, the elbow joint counterbalanceis the mirror image of the shoulder joint counterbalance.

A System that Uses Weight to Store Energy

Here is an example of a system that would be gravity counterbalancedwith a counterbalanced adjustment. It uses weight rather than springs tostore energy. It may not make financial sense, but it is easier tounderstand.

Assume that there is an elevator with only a ground floor and a secondfloor. It might be in a tall tower, with the second floor far above theground floor. A cable pulls the cabin of the elevator up and down. Thecable goes from the top of the cabin, up to a drum, around the drumseveral times, and then back down to a counterweight.

If the counterweight is equal to the weight of the cabin, then thesystem will be balanced. (assuming that the cable is weightless)Assuming no friction, the elevator can go up and down without anyenergy. If passengers get into the cab at the ground floor, energy willbe needed to take them to the top. When they get back on the elevator toreturn to the ground floor, energy will exit the system. Either theenergy will be turned into heat, or a motor-generator may generateelectricity.

The elevator can be adjustably counterbalanced in the following way. Awater tank can be added to the counterweight. Another water tank, fullof water, can be put on the second floor. A third water tank can belocated at the ground floor. Assume that the empty elevator is balancedwhen the tank on the counterweight is empty.

Now, when people get onto the elevator at the ground floor, an equalweight in water can be transferred from the tank on the second floorinto the counterweight tank. The system will be balanced so that theelevator can take the passengers up to the second floor without anyenergy. Only a small motor would be needed to drive the system. If theelevator makes any moves up or down, water should be transferred to orfrom the ground or second floor tanks so that the system is balancedbefore the elevator moves. The elevator can move people and freight upto the second floor until the second floor tank is empty. At that point,someone or something needs to return to ground floor before anythingelse can go up.

The Effect of System Efficiency on Energy Consumption

A purely mechanical, regenerative spring system can be much moreefficient than an electromechanical regenerative system. For example, acar with regenerative braking may have a motor-generator efficiency ofabout 90% and a DC to DC converter efficiency of about 90%. Whenregenerative braking is applied, 90% of the kinetic energy is convertedto electrical energy by the motor-generator. Ten percent of theelectrical energy is lost when it the voltage is converted to a highervoltage by the DC to DC converter. The electrical energy may be storedin a battery or capacitor. When the car accelerates again, 10% of theelectrical energy is lost in the DC to DC converter. The motor-generatorconverts 90% of the remaining electrical energy back into kineticenergy.

The amount of energy that is converted back into kinetic energy afterone cycle can be calculated by taking 0.9 to the 4^(th) power. This isabout 0.65, or 65%. For each braking and acceleration cycle, about 65%of the energy is recovered. This doesn't account for any battery loss. Aspring system should be able to recoup about 95% of the energy percycle.

Let's look at what happens after multiple cycles with each system. Aftertwo cycles with the electromechanical system, 42% of the energy remains.(0.65)²=0.4225

With the spring system, it takes 17 cycles before the remaining energyis down to 42%.(0.95)¹⁷=0.4181

The spring system can do about 17/2 cycles=8.2 times as many cyclesbefore using up about 1−0.42=0.58 or 58% of the available energy.

Looking at it another way, outside energy can be added during each cycleto bring the total kinetic energy back up to 100% of the originalamount. The electromechanical system will need 35% of the originalamount and the spring system will need 5% of the original amount. Thespring system will need only 1/7 or 14% as much energy as theelectromechanical system.

The third spring in the counterbalances mechanism operates as follows:

The first spring mechanism counterbalances the gravity torque on thelink.

The second spring mechanism counterbalances the force needed to adjustthe first spring mechanism. Unfortunately, the first and second springmechanisms are matched at only one link angle. If the system tries toadjust for a different payload at a different link angle, one of twothings will happen. Either extra energy will be needed to make theadjustment, or spring energy will be lost while the adjustment is made.

The third spring mechanism is used to adjust the force of the secondspring mechanism so that it matches the force of the first springmechanism. This enables the link to change payloads at any angle. Energywill not be needed or lost in the process.

In FIGS. 54 a, 54 b and 54 c, I show that a second motor-encoder-brakecan be used to adjust the spring #2-spring #3 combination. The motor canbe replaced with a Scotch Yoke mechanism that is coupled to the axis ofrotation of the link. The spring #2-spring #3 combination would beautomatically adjusted as the link rotates.

The load compensation adjustment of dual capstan #2 in FIGS. 28 a and 28b, can be coupled to the spring carriage.

14. Other Mechanisms to Use in Place of the Spiral Pulley and CableMechanism

Direct Substitution

-   1. A band can be substituted for the cable. The pulley should have a    flat or slightly crowned profile. The resulting mechanism should    have very high efficiency.-   2. A flat belt can be substituted for the cable. The pulley should    have a flat or slightly crowned profile.-   3. A toothed belt or timing belt can be substituted for the cable.    The pulley should have mating grooves.-   4. A roller or silent chain can be substituted for the cable. The    spiral pulley should be replaced with a spiral sprocket with a    mating tooth profile. The sprocket on a chain drive may be smaller    than the corresponding spiral pulley. The chain doesn't have the    same bend radius limitation that a cable does. The chain drive will    have a chordal error.    Other Variable Ratio Transmissions-   5. The spiral pulley and cable mechanism is a type of variable ratio    transmission. Gears can be made with a spiral profile. Gears with    mating spiral profiles can be connected to various types of springs.    A torsion spring can be connected directly to a spiral gear. The    mating spiral gear can be connected to another torsion spring. If it    is difficult to achieve the desired ratio with one pair of gears,    two pairs can be used in series. This is very similar to the dual    spiral pulley system.-   6. A cam with a roller follower can be used in place of the spiral    pulley and cable. FIGS. 29 a and b show cam and roller mechanisms    connected to a spring. The cam in FIG. 29 a is an external cam. The    roller follower rolls on the outside surface of the cam. The cam in    FIG. 29 b is an internal cam. The roller follower rolls on the    inside surface of the cam. The cam profiles can be generated using a    method similar to the way that the spiral pulley profiles were    generated. For example, the cam can be designed to produce a    constant torque over its range of rotation. Differential equations    can be generated and solve numerically for the required cam profile.    As with the spiral pulleys, the torque profile does not need to be    constant, parabolic or other profiles should work too. The cam and    roller mechanism should be very efficient, and it may be smaller    than a spiral pulley with the same output.    15. Other Mechanisms Constant Force or Torque Mechanism

FIGS. 34 a and 34 b show an adjustable constant force mechanism. It issimilar to the adjustable translational counterbalance mechanisms shownin FIGS. 35 a, 35 b, 36 a, 36 b, 37 a, and 37 b. In FIGS. 37 a and 37 b,cable #3 is attached to a pivot that translates with the carriage. Cable#3 wraps around a sliding pivot spiral pulley. The other spiral pulleyin FIGS. 37 a and 37 b is a fixed pivot spiral pulley.

In FIGS. 34 a and 34 b, both of the spiral pulleys are fixed pivotspiral pulleys. The torque on the triple capstan is constant. The torquecan be adjusted by changing the tension on either of the springs. Cable#3 and the third capstan convert the torque to a constant force. Theforce from cable #3 can be used to counterbalance a translating link.

Adjustable Stiffness Rotary Counterbalance

FIGS. 55 a and 55 b show an adjustable rotary counterbalance mechanism.Looking back at equation 6, the magnitude of the counterbalance torquecan be changed by adjusting dimension a, dimension b, or spring constantK₁. In most of the previous rotary counterbalance mechanisms, dimensiona was adjusted. In FIGS. 55 a and 55 b, the spring constant is adjusted.

The mechanism shown below the idler pulley is a constant torquemechanism. It's the same mechanism shown in FIGS. 34 a and 34 b. Spiralpulley #3 has been rigidly attached to the dual capstan. Spiral pulley#3 is a fixed pivot spiral pulley. It's the type of spiral pulley thatconverts the force from a linear spring into a constant torque. In thiscase, the torque from the lower mechanism is converted to a force incable #3. The effective stiffness of the force in cable #3 can bechanged by adjusting the lower constant torque mechanism.

Moving the Counterbalance Mechanism Away from the Link

The counterbalance mechanism shown in FIGS. 55 a and 55 b can be locatedaway from the link that's being counterbalanced. A pantograph mechanismcan also allow a counterbalance mechanism to be located away from thelink. In the FIGS. 55 a and 55 b mechanism, only one cable is needed totransmit the force to the link. The counterbalance mechanism can belocated on an adjacent link by routing cable #3 along the connectingaxis of rotation. The inertia of the arm can be reduced by moving themass of the mechanism to a link that's closer to ground. Less torque mayalso be needed to support the weight of the mechanism.

The Scotch Yoke counterbalance shown in FIGS. 41 a and 41 b can also belocated away from the link. FIGS. 46 a and 46 b show how a single cablecan be used to couple the output of a constant force mechanism to theScotch Yoke mechanism.

Some Uses and Advantages of The Adjustable Load, Energy ConservingCounterbalance Mechanism And the Multiple Serial Link Balance Mechanisms

Robotics

Advantages

-   -   Much larger payloads can be lifted with the same motors.    -   Smaller less expensive motors can be used. The power supply and        electronics can be smaller too.    -   Less energy consumption. It will be a big advantage for mobile        robots.    -   Able to apply a constant force in any direction without any        energy consumption. Infinitely more efficient than with servo        motors.    -   Multiple link mechanisms enable counterbalancing of arms with        more degrees of freedom than currently possible.    -   With a counterbalance, the motors don't have to hold the weight        of the payload. Accuracy and response or bandwidth is improved.        Accurate and delicate movement is possible, even with a heavy        load.    -   Counterbalancing with weights increases the weight and inertia        of the system. Counterbalancing with springs will add very        little to the weight and inertia. With less inertia, the robot        can accelerate faster    -   Zero stiffness will make it safer around humans.

The foregoing description of preferred embodiments of the presentinvention has been provided for the purposes of illustration anddescription. It is not intended to be exhaustive or to limit theinvention to the precise forms disclosed. Obviously, many modificationsand variations will be apparent to the practitioner skilled in the art.The embodiments were chosen and described in order to best explain theprinciples of the invention and its practical application, therebyenabling others skilled in the art to understand the invention forvarious embodiments and with various modifications that are suited tothe particular use contemplated. It is intended that the scope of theinvention be defined by the following claims and their equivalence.

1. A mechanical arm, comprising: a forearm; a first spring; an upper armconnected between the forearm and the first spring, wherein the forearmapplies a mass that generates a first moment at the upper arm; a copydevice associated with the upper arm, the copy device copying the firstmoment to the first spring, wherein the first spring applies acounter-force to a tension member connected between the first spring andthe upper arm to resist at least a portion of the moment; a shoulderwith which the upper arm is rotatably connected, wherein the upper armgenerates a second moment at a junction of the upper arm and theshoulder; a second spring, wherein the second spring applies acounter-force to resist at least a portion of the second moment; a pitchgimbal mechanism pivotable based on pitch movement of the upper arm atthe shoulder; and a yaw gimbal mechanism pivotable based on yaw movementof the upper arm at the shoulder.
 2. The mechanical arm of claim 1,wherein: the upper arm is adapted to lift a payload, whereby the firstmoment is increased by the payload; and the first spring is adjustableto adjustably apply a counter-force to balance the first moment.
 3. Themechanical arm of claim 2, wherein the first spring is adjustable toadjustably apply a counter-force to balance the first moment when thefirst moment is increased or decreased.
 4. The mechanical arm of claim2, further comprising: a first spring adjustment device to alter aresponse characteristic of the first spring; and wherein the firstmoment is selectively balanced by adjusting the first spring with thefirst spring adjustment device.
 5. The mechanical arm of claim 4,wherein: the first spring adjustment device is a first movable carriage;the first spring is fixedly connected with the first movable carriage;and a stiffness of the first spring changes with a position of the firstmovable carriage.
 6. The mechanical arm of claim 1, wherein: one or bothof the upper arm and the forearm is adapted to lift a payload, wherebythe second moment is increased by the payload; the second spring isadjustable to adjustably apply a counter-force to balance the secondmoment.
 7. The mechanical arm of claim 6, wherein the second spring isadjustable to adjustably apply a counter-force to balance the secondmoment when the second moment is increased or decreased.
 8. Themechanical arm of claim 6, further comprising: a second springadjustment device to alter a response characteristic of the secondspring; and wherein the second moment is selectively balanced byadjusting the second spring with the second spring adjustment device. 9.The mechanical arm of claim 8, wherein: the second spring adjustmentdevice is a second movable carriage; the second spring is fixedlyconnected with the second movable carriage; and a stiffness of thesecond spring changes with a position of the second movable carriage.10. The mechanical arm of claim 1, further comprising: a pitch gimbalmechanism to adjust a connection of the tension member to the firstspring based on pitch movement of the forearm at the upper arm.
 11. Amechanical arm comprising: a forearm adapted to support a payload; anupper arm rotatably connected with the forearm, wherein a first momentis generated at a connection of the forearm and the upper arm; whereinthe forearm is moveable in a yaw motion or a pitch motion relative tothe upper arm; a first spring; a link rotatably connected with the upperarm at a first end and connected to the first spring at a second end bya tension member; a copy device copying the first moment to the link,wherein the first spring applies a first counter-force to the tensionmember to resist at least a portion of the first moment copied to thelink; wherein the link is moveable in a yaw motion or a pitch motionrelative to the upper arm in response to a corresponding yaw motion orpitch motion of the forearm; a shoulder with which the upper arm isrotatably connected, wherein the upper arm generates a second moment ata junction of the upper arm and the shoulder; a second spring, whereinthe second spring applies a second counter-force to resist at least aportion of the second moment; a first cable gimbal to adjust anorientation of the first tension member relative to the first springbased on one or both of yaw movement and pitch movement of the link; anda second cable gimbal pivotable relative to the second spring based onpitch movement of the upper arm at the shoulder.
 12. The mechanical armof claim 11, wherein the first spring is adjustable to adjustably applya counter-force to balance the first moment when the first moment isincreased or decreased.
 13. The mechanical arm of claim 12, furthercomprising: a first spring adjustment device to alter a responsecharacteristic of the first spring; and wherein the first moment isselectively balanced by adjusting the first spring with the first springadjustment device.
 14. The mechanical arm of claim 13, wherein: thefirst spring adjustment device is a first movable carriage; the firstspring is fixedly connected with the first movable carriage; and astiffness of the first spring changes with a position of the firstmovable carriage.
 15. The mechanical arm of claim 11, wherein the secondspring is adjustable to adjustably apply a counter-force to balance thesecond moment when the second moment is increased or decreased.
 16. Themechanical arm of claim 15, further comprising: a second springadjustment device to alter a response characteristic of the secondspring; and wherein the second moment is selectively balanced byadjusting the second spring with the second spring adjustment device.17. The mechanical arm of claim 16, wherein: the second springadjustment device is a second movable carriage; the second spring isfixedly connected with the second movable carriage; and a stiffness ofthe second spring changes with a position of the second movablecarriage.